aa r X i v : . [ m a t h . A C ] J a n A New Type of Bases for Zero-dimensional Ideals
Sheng-Ming Ma
Abstract
We formulate a substantial improvement on Buchberger’s algorithm forGr¨obner bases of zero-dimensional ideals. The improvement scales downthe phenomenon of intermediate expression swell as well as the complexityof Buchberger’s algorithm to a significant degree. The idea is to computea new type of bases over principal ideal rings instead of over fields likeGr¨obner bases. The generalizations of Buchberger’s algorithm from overfields to over rings are abundant in the literature. However they are limitedto either computations of strong Gr¨obner bases or modular computationsof the numeral coefficients of ideal bases with no essential improvement onthe algorithmic complexity. In this paper we make pseudo-divisions withmultipliers to enhance computational efficiency. In particular, we developa new methodology in determining the authenticity of the factors of thepseudo-eliminant, i.e., we compare the factors with the multipliers of thepseudo-divisions instead of the leading coefficients of the basis elements. Inorder to find out the exact form of the eliminant, we contrive a modularalgorithm of proper divisions over principal quotient rings with zero divisors.The pseudo-eliminant and proper eliminants and their corresponding basesconstitute a decomposition of the original ideal. In order to address theideal membership problem, we elaborate on various characterizations ofthe new type of bases. In the complexity analysis we devise a scenariolinking the rampant intermediate coefficient swell to B´ezout coefficients,partially unveiling the mystery of hight-level complexity associated with thecomputation of Gr¨obner bases. Finally we make exemplary computationsto demonstrate the conspicuous difference between Gr¨obner bases and thenew type of bases.
Contents
Email: [email protected]; [email protected]
Address:
School of Mathematics, Peking University of Hang Kong, Beijing, China.2020
Mathematics Subject Classification.
Key words and phrases: zero-dimensional ideal, eliminant, ideal bases, complexity, Gr¨obnerbases, pseudo-division, modular method, intermediate coefficient swell, principal ideal rings. Pseudo-eliminant Divisors and Compatibility 135 Analysis of Incompatible Divisors via Modular Method 216 A New Type of Bases for Zero-dimensional Ideals 417 Further Improvements on the Algorithm 568 Complexity Comparison with Gr¨obner Bases 589 Examples and Paradigmatic Computations 6210 Conclusion and Remarks 71
After Buchberger initiated his celebrated algorithm in his remarkable PhD thesis[Buc65], the theory of Gr¨obner bases has been established as a standard tool inalgebraic geometry and computer algebra, yielding algorithmic solutions to manysignificant problems in mathematics, science and engineering [BW98]. As a result,there have been many excellent textbooks on the subject such as [AL94] [BW93][CLO15] [KR00] [DL06] [GP08] [EH12] [GG13].Nonetheless the computational complexity of Gr¨obner bases often demandsan enormous amount of computing time and storage space even for problemsof moderate sizes, which severely impedes its practicality and dissemination. Astriking phenomenon is in the computation of Gr¨obner bases over the rational fieldwith respect to the lex ordering, when the coefficients of the final basis elementsswell to extremely complicated rational numerals even though the coefficients ofthe original ideal generators are quite moderate. Example 9.2 in this paper shouldbe impressive enough to illustrate such a phenomenon. An even more dramaticphenomenon is the “intermediate expression swell” referring to a generation of ahuge number of intermediate polynomials with much more gigantic coefficients andlarger exponents than those of the final basis elements during the implementationof the classical algorithm. In [CLO15, P116] and [GG13, P616, 21.7] there are somebrief reviews on the complexity issues associated with the classical algorithm.These challenges have stimulated decades of ardent endeavors in improvingthe efficiency of the classical algorithm. The methodologies such as the normalselection strategies and signatures effectively diminish the number of intermediatepolynomials spawned during the process of algorithmic implementations [Buc85][GMN91] [BW93, P222, 5.5] [Fau02] [EF17]. The modular and p -adic techniquesbased on “lucky primes” and Hensel lifting have been adopted to control therampant growth of the intermediate coefficients albeit being limited to numeralcoefficients only [Ebe83] [Win87] [ST89] [Tra89] [Pau92] [Gra93] [Arn03]. Thereare also Gr¨obner basis conversion methods such as the FGLM algorithm [FGL93]and Gr¨obner Walk [CKM97], a detailed description of which can be found in[CLO05, P49, §
3; P436, §
5] and [Stu95]. The idea of these methods is to computeanother Gr¨obner basis with respect to a different but favorable monomial orderingbefore converting it to the desired Gr¨obner basis. Albeit with all these endeavors2ver the decades, the high-level complexity associated with the Gr¨obner basiscomputations remains a conundrum.Another train of thoughts over the past decades is to generalize Gr¨obner basesfrom over fields to over rings. Among the copious and disparate coefficient ringsthat we shall not enumerate here, the Gr¨obner bases over principal ideal rings arepertinent to the new type of bases in this paper. There is an excellent exposition onGr¨obner bases over rings and especially over PIDs in [AL94, Chapter 4]. Howeverthe focal point of the exposition is on the strong Gr¨obner bases that resemble theGr¨obner bases over fields and hence are still plagued with complexity problems.In this paper we take a novel approach by defining a new type of bases overprincipal quotient rings instead of over numeral fields like Gr¨obner bases. It is anatural approach since for a zero-dimensional ideal, the final eliminant is always aunivariate polynomial after eliminating all the other variables. With the principalideals generated by the eliminant factors serving as moduli, we obtain an elegantdecomposition of the original ideal into pairwise relatively prime ideals. We alsouse pseudo-divisions and multipliers to enhance computational efficiency. In theexemplary computations in Section 9, it is conspicuous that this new approachscales down both the high-level complexity and gigantic numeral coefficients ofthe Gr¨obner bases over rational fields.In practice the Wu’s method [Wu83] is more commonly used than the Gr¨obnerbasis method since it is based on pseudo-divisions and thus more efficient. Howeverthe pseudo-divisions adopted by Wu’s method usually lose too much algebraicinformation of the original ideals. In Section 2 we recall some rudimentary facts onmonomial orderings and then define pseudo-divisions over PIDs. The multipliersfor the pseudo-divisions in this paper are always univariate polynomials so as toavoid losing too much algebraic information of the original ideals. The pseudo-divisions of this ilk also dispose of the solvability condition for the linear equationsof leading coefficients imposed by the classical division algorithm over rings. Pleaserefer to Remark 2.9 for details.Algorithm 3.9 is one of the pivotal algorithms in the paper. It computes thepseudo-eliminant and pseudo-basis as per the elimination ordering in Definition3.1. The purpose of Corollary 3.7 and Lemma 3.8 is to trim down the number of S -polynomials to be pseudo-reduced. They are highly effective in this respect asillustrated by the exemplary computations in Section 9.The pseudo-eliminant might contain factors that are not the bona fide ones ofthe eliminant. The discrimination among these factors for authenticity is basedon a crucial methodology, i.e., the pseudo-eliminant should be compared with themultipliers of the pseudo-divisions instead of the leading coefficients of the basiselements. Example 4.12 shows that the multipliers of the pseudo-divisions aremore reliable than the leading coefficients of the basis elements. The multipliermethodology is incorporated into Theorem 4.10 establishing that the compatiblepart of the pseudo-eliminant constitutes a bona fide factor of the eliminant. Thisis one of the primary conclusions of the paper. The multiplier and its property inLemma 4.8 generalize the syzygy theory over fields and PIDs for Gr¨obner bases andis another substantiation of the multiplier methodology. Please refer to Remark 4.9for the comment. The compatible and incompatible parts of the pseudo-eliminantare defined in Definition 4.5 and computed via Algorithm 4.6. In particular, weobtain a squarefree decomposition of the incompatible part ip ( χ ε ) by Algorithm 4.63ia a univariate squarefree factorization of the pseudo-eliminant χ ε by Algorithm4.2. We avoid a complete univariate factorization of the pseudo-eliminant χ ε dueto the concerns on computational complexity.We conduct a complete analysis of the incompatible part ip ( χ ε ) of the pseudo-eliminant χ ε in Section 5 based on modular algorithms with the composite divisorsobtained in Algorithm 4.6 as the moduli. The advantages are that we have one lessvariable than the classical algorithm and the composite divisors are usually smallpolynomial factors of the pseudo-eliminant χ ε . However the disadvantage is thatthe computations are over the principal quotient rings (PQR) that might containzero divisors. As a result, we redefine S -polynomials in Definition 5.11 carefullyin order to obviate the zero multipliers incurred by the least common multiple ofleading coefficients. Algorithm 5.20 is pivotal in procuring the proper eliminantsand proper bases by proper divisions as in Theorem 5.10. We prove rigorously inTheorem 5.26 that the nontrivial proper eliminants obtained in Algorithm 5.20are de facto the bone fide factors of the eliminant of the original ideal. Themeticulous arguments in this primary conclusion are to ensure that our argumentsare legitimate within the algebra R q [ ˜ x ] that contains zero divisors. Similar toLemma 4.8, we generalize the classical syzygy theory over fields and PIDs forGr¨obner bases to the one in Lemma 5.25. Further, we also have Corollary 5.14and Lemma 5.18 to trim down the number of S -polynomials for proper reductions.We render two equivalent characterizations on the pseudo-bases B ε obtained inAlgorithm 3.9. The first characterization is the identity (6.15) in terms of leadingterms whereas the second one is Theorem 6.13 via gcd -reductions as defined inTheorem 6.12. We have the same kind of characterizations in Theorem 6.15 forthe proper bases B q and B p obtained in Algorithm 5.20. These bases as in (6.35)correspond to a decomposition of the original ideal in (6.2) whose modular versionis in (6.36). We can define a unique normal form of a polynomial in R q [ ˜ x ] withrespect to the original ideal by the Chinese Remainder Theorem as in Lemma6.20 since the ideal decomposition in (6.2) is pairwise relatively prime. In theremaining part of this section we define irredundant, minimal and reduced basesthat possess different levels of uniqueness.In Section 7 we make some further improvements on the algorithms by Principle7.1. The highlight of Section 8 is Lemma 8.3 in which we contrive a specialscenario consisting of two basis elements, a detailed analysis of which reveals thatthe classical algorithm contains the Euclidean algorithm computing the greatestcommon divisor of the leading coefficients. Moreover, the results in (8.12) and(8.13) contain the B´ezout coefficients u and v that might swell to an enormoussize like in Example 8.1. By contrast the computation of our new type of S -polynomial as in (8.14) yields the above results in one step without the B´ezoutcoefficients. This might help to unveil the mystery of intermediate coefficient swellas well as high-level complexity associated with the Gr¨obner basis computations.We make two exemplary computations in Section 9 with the second one beingmore sophisticated than the first one. It contains a paradigmatic computationof proper eliminants and proper bases over principal quotient rings with zerodivisors as in Algorithm 5.20. We provide a detailed explanation for each step ofthe computation to elucidate the ideas of this new type of bases.As usual, we denote the sets of complex, real, rational, integral and naturalnumbers as C , R , Q , Z and N respectively. In this paper, we use the following4otations for a ring R : R ∗ := R \ { } ; R × denotes the set of units in R . With x = ( x , . . . , x n ) and α = ( α , . . . , α n ), we denote a monomial x α · · · x α n n as x α and a term as c x α with the coefficient c ∈ R ∗ . We also use the boldface x toabbreviate the algebra R [ x , . . . , x n ] over a ring R as R [ x ]. The notation h A i denotes an ideal generated by a nonempty subset A ⊂ R [ x ]. Further, K usuallydenotes a perfect field that is not necessarily algebraically closed unless specified.In most cases we treat the algebra K [ x ] as ( K [ x ])[ ˜ x ] over the ring R = K [ x ]with the variables ˜ x := ( x , . . . , x n ). In this article we adopt a pseudo-divison of polynomials over a principal idealdomain as in Theorem 2.7. We shall abbreviate a principal ideal domain as a PIDand denote it as R henceforth.Let R be a PID and R [ x ] a polynomial algebra R [ x , . . . , x n ] over R . Let usdenote the set of monomials in x = ( x , . . . , x n ) as [ x ] := { x α : α ∈ N n } . Anonzero ideal I ⊂ R [ x ] is called a monomial ideal if I is generated by monomialsin [ x ] . By Hilbert Basis Theorem we can infer that every monomial ideal in R [ x ]is finitely generated since a PID R is Noetherian. Lemma 2.1.
Let R be a PID. Consider a monomial ideal I = h x α : α ∈ E i in R [ x ] with E ⊂ N n \ { } . We have the following conclusions:(i) A term c x β ∈ I for c ∈ R ∗ if and only if there exists an α ∈ E such that x β is divisible by x α ;(ii) A polynomial f ∈ I if and only if every term of f lies in I .Proof. It suffices to prove the necessity of the two conclusions. Suppose that g = P sj =1 q j x α j with g representing the term c x β ∈ I as in (i), or f ∈ I as in (ii).Here q j ∈ R [ x ] and α j ∈ E for 1 ≤ j ≤ s . We expand each q j into individual termsand compare those with the same multi-degrees on both sides of the equality. Theconclusion readily follows since every term on the right hand side of the equalityis divisible by some x α j with α j ∈ E .A total ordering ≻ on the monomial set [ x ] satisfies that for every pair x α , x β ∈ [ x ] , exactly one of the following relations holds: x α ≻ x β , x α = x β , or x α ≺ x β .Moreover, x α (cid:23) x β means either x α ≻ x β or x α = x β . A well-ordering ≻ on [ x ] satisfies that every nonempty subset A ⊂ [ x ] has a minimal element. Thatis, there exists x α ∈ A such that x β (cid:23) x α for every x β ∈ A . A well-ordered setis always a totally ordered set since every subset consisting of two elements has aminimal element.It is evident that under a well-ordering ≻ on [ x ] , there is no infinite strictlydecreasing sequence x α ≻ x α ≻ x α ≻ · · · in [ x ] (or every strictly decreasingsequence in [ x ] terminates). Nonetheless we have a much easier description asfollows under the Noetherian condition. 5 roposition 2.2. Let ≻ be a total ordering on [ x ] such that x α · x γ ≻ x β · x γ when x α ≻ x β for all x α , x β , x γ ∈ [ x ] . Then ≻ is a well-ordering on [ x ] if andonly if x γ (cid:23) for all γ ∈ N n .Proof. Suppose that ≻ is a well-ordering. Then [ x ] has the smallest element whichwe denot as x β . If 1 ≻ x β , then x β ≻ x β , contradicting the minimality of x β .Hence follows the necessity of the conclusion.Suppose that A ⊂ [ x ] is nonempty. To prove the sufficiency, it suffices to provethat A has a minimal element in terms of the ordering ≻ . Let K be a nontrivialfield and h A i := h{ x α : α ∈ A }i the monomial ideal generated by A in K [ x ]. Asper Hilbert Basis Theorem, h A i has a finite basis as h A i = h x α , x α , . . . , x α s i .Since ≻ is a total ordering, we relabel the subscripts such that x α s ≻ · · · ≻ x α .Now x α is the minimal element of A . In fact, for every x α ∈ h A i , according toLemma 2.1, x α is divisible by one of { x α j } for 1 ≤ j ≤ s . Assume that x α isdivisible by x α j , i.e., x α = x α j · x γ for some γ ∈ N n . Then x γ (cid:23) x α (cid:23) x α j (cid:23) x α . Definition 2.3. A monomial ordering on [ x ] is a well-ordering on [ x ] such that x α · x γ ≻ x β · x γ when x α ≻ x β for all x α , x β , x γ ∈ [ x ] . In particular, we have x γ (cid:23) γ ∈ N n . Notation . Let R be a PID and f = P α c α x α a polynomial in R [ x ]. Let ≻ bea monomial ordering. We denote the support of f as supp( f ) := { x α ∈ [ x ] : c α =0 } ⊂ [ x ] . In particular, we define supp( f ) := { } when f ∈ R ∗ and supp( f ) := ∅ when f = 0.If f has a term c β x β that satisfies x β := max ≻ { x α ∈ supp( f ) } , then weuse the following terminologies hereafter. The leading term of f is denoted as lt ( f ) := c β x β ; The leading monomial of f is denoted as lm ( f ) := x β ; The leading coefficient of f is denoted as lc ( f ) := c β ∈ R ∗ .Let B = { b j : 1 ≤ j ≤ s } be a polynomial set in R [ x ] \ { } . We denote theleading monomial set { lm ( b j ) : 1 ≤ j ≤ s } as lm ( B ). Let us also denote themonomial ideal generated by lm ( B ) in R [ x ] as h lm ( B ) i .In what follows we use gcd( a, b ) and lcm( a, b ) to denote the greatest commondivisor and least common multiple of a, b ∈ R ∗ respectively over a PID R . Definition 2.5 (Term pseudo-reduction in R [ x ] over a PID R ) . Let R be a PID and ≻ a monomial ordering on [ x ] . For f ∈ R [ x ] \ R and g ∈ R [ x ] \ { } , suppose that f has a term c α x α such that x α ∈ supp( f ) ∩ h lm ( g ) i .Then we can make a pseudo-reduction of the term c α x α of f by g as follows. h = µf − m x α lt ( g ) g (2.1)with the multipliers m := lcm( c α , lc ( g )) and µ := m/c α ∈ R ∗ . We call h the remainder of the pseudo-reduction and µ the interim multiplier on f with respectto g . It is also called the “leading power product” in the literature. Here we adopt the conventionthat is consistent with the terminology of “ monomial ideals”. efinition 2.6 (Pseudo-reduced polynomial) . Let R be a PID and ≻ a monomial ordering on [ x ] . A polynomial r ∈ R [ x ] is pseudo-reduced with respect to a polynomial set B = { b j : 1 ≤ j ≤ s } ⊂ R [ x ] \ R if supp( r ) ∩ h lm ( B ) i = ∅ . In particular, this includes the special case when r = 0and hence supp( r ) = ∅ . We also say that r is pseudo-reducible with respect to B if it is not pseudo-reduced with respect to B , i.e., supp( r ) ∩ h lm ( B ) i 6 = ∅ . Theorem 2.7 (Pseudo-division in R [ x ] over a PID R ) . Let R be a PID and ≻ a monomial ordering on [ x ] . Suppose that B = { b j : 1 ≤ j ≤ s } ⊂ R [ x ] \ R is a polynomial set. For every f ∈ R [ x ] , there exist a multiplier λ ∈ R ∗ as well as a remainder r ∈ R [ x ] and quotients q j ∈ R [ x ] for ≤ j ≤ s such that λf = s X j =1 q j b j + r, (2.2) where r is pseudo-reduced with respect to G . Moreover, the polynomials in (2.2) satisfy the following condition: lm ( f ) = max (cid:8) max ≤ j ≤ s { lm ( q j b j ) } , lm ( r ) (cid:9) . (2.3) Proof. If f is already pseudo-reduced with respect to B , we just take r = f and q j = 0 for 1 ≤ j ≤ s . Otherwise we define x α := max ≻ { supp( f ) ∩ h lm ( B ) i} .There exists some j such that x α is divisible by lm ( b j ) as per Lemma 2.1 (i).We make a pseudo-reduction of the term c α x α of f by b j in the same way as theterm pseudo-reduction in (2.1). We denote the remainder also as h and x β :=max ≻ { supp( h ) ∩ h lm ( B ) i} if h is not pseudo-reduced with respect to B . It iseasy to see that x α ≻ x β after the above term pseudo-reduction. We repeat suchterm pseudo-reductions until the remainder h is pseudo-reduced with respect to B . Since the monomial ordering ≻ is a well-ordering by Definition 2.3, the termpseudo-reductions terminate in finite steps. Hence follows the representation (2.2)in which the multiplier λ ∈ R ∗ is a product of such interim multipliers µ as in(2.1).To prove the equality in (2.3), it suffices to prove it for the term pseudo-reduction in (2.1). In fact, the pseudo-division in (2.2) is just a composition ofthe term pseudo-reductions in (2.1) and the remainder h in (2.1) shall eventuallybecome the remainder r in (2.2). In (2.1) the leading monomial of m x α g/ lt ( g )is x α . Hence either lm ( f ) = x α , or lm ( f ) ≻ x α in which case lm ( f ) = lm ( h ) in(2.1). Thus follows the equality in (2.3). Definition 2.8.
Let R be a PID and f ∈ R [ x ]. Suppose that B = { b j : 1 ≤ j ≤ s } ⊂ R [ x ] \ R is a polynomial set over R . We call the expression in (2.2) a pseudo-division of f by B . More specifically, we name the polynomial r in (2.2)as a remainder of f on pseudo-division by B and λ ∈ R ∗ in (2.2) a multiplier of the pseudo-division. We say that f pseudo-reduces to the remainder r via the multiplier λ ∈ R ∗ modulo B . We also call it a pseudo-reduction of f by B .The proof of Theorem 2.7 shows that the multiplier λ in (2.2) is a finite productof the interim multipliers µ as in (2.1). Based on the proof of Theorem 2.7 we caneasily contrive a pseudo-division algorithm. We do not elaborate on it here sinceit is quite straightforward. 7 emark . There is a difference between the above pseudo-division algorithmand the traditional division algorithm over a PID. In the traditional one as in[AL94, P207, Algorithm 4.1.1], it is required that the linear equation lc ( f ) = P sj =1 b j · lc ( f j ) as in [AL94, P204, (4.1.1)] be solvable for b j ’s over R . The pseudo-division algorithm does not have this extra requirement. Their major differenceis the multiplier λ ∈ R ∗ in (2.2). Let K be a field and x denote variables ( x , . . . , x n ) as before. In this sectionlet us consider the case when the PID R in Section 2 bears the particular form R = K [ x ] with x being the first variable of x . In this case the polynomials inthe algebra K [ x ] over K can be viewed as those in ( K [ x ])[ ˜ x ] over K [ x ] withthe variables ˜ x = ( x , . . . , x n ) and coefficients in K [ x ]. It is evident that thepseudo-division of polynomials in Theorem 2.7 applies here without any essentialchange.Unless specified, in what follows we shall always treat the algebra K [ x ] over K as the algebra ( K [ x ])[ ˜ x ] over K [ x ]. Hence for f ∈ ( K [ x ])[ ˜ x ] in this section,its leading coefficient lc ( f ) and leading monomial lm ( f ) in Notation 2.4 nowsatisfy lc ( f ) ∈ ( K [ x ]) ∗ and lm ( f ) ∈ [ ˜ x ] respectively. Here [ ˜ x ] denotes the setof nonzero monomials in the variables ˜ x = ( x , . . . , x n ). Moreover, we use ( f )to denote a principal ideal in K [ x ] generated by f ∈ K [ x ]. Recall that h f i denotes a principal ideal in ( K [ x ])[ ˜ x ] = K [ x ] generated by either f ∈ K [ x ] or f ∈ ( K [ x ])[ ˜ x ]. Definition 3.1 (Elimination ordering on ( K [ x ])[ ˜ x ]) . An elimination ordering on ( K [ x ])[ ˜ x ] is a monomial ordering on [ x ] such thatthe ˜ x variables are always larger than the x variable. That is, x α ˜ x γ ≻ x β ˜ x δ ifand only if ˜ x γ ≻ ˜ x δ or, ˜ x γ = ˜ x δ and α > β .In what follows let us suppose that I is a zero-dimensional ideal of K [ x ] =( K [ x ])[ ˜ x ]. We have the following well-known conclusion. Proposition 3.2.
For a zero-dimensional ideal I ⊂ ( K [ x ])[ ˜ x ] , we always have I ∩ K [ x ] = { } .Proof. Please refer to [BW93, P272, Lemma 6.50] or [KR00, P243, Proposition3.7.1(c)].
Definition 3.3 (Eliminant) . For a zero-dimensional ideal I ⊂ ( K [ x ])[ ˜ x ], the principal ideal I ∩ K [ x ] in K [ x ] is called the elimination ideal of I . Let us denote its generator as χ thatsatisfies I ∩ K [ x ] = ( χ ) being a principal ideal in K [ x ]. We call χ the eliminant of the zero-dimensional ideal I henceforth.In what follows let us elaborate on a revised version of Buchberger’s algorithm.The purpose is to compute not only a pseudo-basis but also a pseudo-eliminant of Please also refer to [AL94, P69, Definition 2.3.1] and [EH12, P33, Definition 3.1]. Please note that a zero-dimensional ideal I is always a proper ideal such that I = K [ x ]. I ∩ K [ x ]. Let us first recall the S -polynomial over a PID asin the following definition . Definition 3.4 ( S -polynomial) . Suppose that f, g ∈ ( K [ x ])[ ˜ x ] \ K [ x ]. Let us denote m := lcm( lc ( f ) , lc ( g )) ∈ K [ x ] and ˜ x γ := lcm( lm ( f ) , lm ( g )) ∈ [ ˜ x ] . Then the polynomial S ( f, g ) := m ˜ x γ lt ( f ) f − m ˜ x γ lt ( g ) g (3.1)is called the S -polynomial of f and g .It is easy to verify that the S -polynomial satisfies the following inequality dueto the cancellation of leading terms in (3.1): lm ( S ( f, g )) ≺ ˜ x γ = lcm( lm ( f ) , lm ( g )) . (3.2)When g ∈ ( K [ x ]) ∗ and f ∈ ( K [ x ])[ ˜ x ] \ K [ x ], we take lm ( g ) = 1 and m = lcm( lc ( f ) , g ). The S -polynomial in (3.1) becomes: S ( f, g ) := m lc ( f ) f − m · lm ( f ) . (3.3) Lemma 3.5.
When g ∈ ( K [ x ]) ∗ and f ∈ ( K [ x ])[ ˜ x ] \ K [ x ] , the S -polynomial in (3.3) satisfies: dS ( f, g ) = ( f − lt ( f )) · g := f g (3.4) with d := gcd( lc ( f ) , g ) ∈ K [ x ] and f := f − lt ( f ) .Proof. Let us denote l f := lc ( f ). Based on the equality m/l f = g/d , the S -polynomial in (3.3) becomes: S ( f, g ) = gfd − gl f d · lm ( f ) = gd ( f − lt ( f )) = f gd . Lemma 3.5 can be generalized to the following conclusion:
Lemma 3.6.
For f, g ∈ ( K [ x ])[ ˜ x ] \ K [ x ] , suppose that lm ( f ) and lm ( g ) arerelatively prime. Let us denote d := gcd( lc ( f ) , lc ( g )) . Then their S -polynomialin (3.1) satisfies: dS ( f, g ) = f g − g f (3.5) with f := f − lt ( f ) and g := g − lt ( g ) . Moreover, we have: lm ( S ( f, g )) = max { lm ( f g ) , lm ( g f ) } . (3.6) Proof. If lm ( f ) and lm ( g ) are relatively prime, then we have an identity ˜ x γ = lm ( f ) · lm ( g ) in (3.1). For convenience, let us denote l f := lc ( f ), l g := lc ( g ) and d := gcd( l f , l g ). Then we have the identities m/l f = l g /d and m/l g = l f /d with m = lcm( l f , l g ) as in (3.1). We substitute these identities into (3.1) to obtain: S ( f, g ) := l g · lm ( g ) d f − l f · lm ( f ) d g = 1 d ( lt ( g ) · f − lt ( f ) · g )= 1 d (( g − g ) f − ( f − f ) g ) = 1 d ( f g − g f ) (3.7)= 1 d ( f ( g + lt ( g )) − g ( f + lt ( f ))) = 1 d ( f · lt ( g ) − g · lt ( f )) . (3.8) Please also refer to [AL94, P249, (4.5.1)] or [BW93, P457, Definition 10.9]. c α ˜ x α of f and everyterm c β ˜ x β of g , we have ˜ x α · lm ( g ) = ˜ x β · lm ( f ) since lm ( g ) and lm ( f ) arerelatively prime and moreover, we have ˜ x α ≺ lm ( f ) and ˜ x β ≺ lm ( g ). As a result,no term of f · lt ( g ) cancels no term of g · lt ( f ) in (3.8). Thus follows the equality(3.6). Corollary 3.7.
Suppose that lm ( f ) and lm ( g ) are relatively prime for f, g ∈ ( K [ x ])[ ˜ x ] \ K [ x ] . Then their S -polynomial S ( f, g ) as in (3.1) can be pseudo-reduced to by f and g as in Theorem 2.7 with the multiplier λ = d and quotients f and g as in (3.5) .In particular, for g ∈ ( K [ x ]) ∗ and f ∈ ( K [ x ])[ ˜ x ] \ K [ x ] , their S -polynomial S ( f, g ) in (3.3) can be pseudo-reduced to by g with the multiplier λ = d andquotient f as in (3.4) .Proof. The conclusions readily follow from Lemma 3.6 and Lemma 3.5 based onTheorem 2.7.For two terms c α ˜ x α , c β ˜ x β ∈ ( K [ x ])[ ˜ x ] with c α , c β ∈ K [ x ], let us denotelcm( c α ˜ x α , c β ˜ x β ) := lcm( c α , c β ) · lcm( ˜ x α , ˜ x β ). Then we have the following notation:lcm( lt ( f ) , lt ( g )) := lcm( lc ( f ) , lc ( g )) · lcm( lm ( f ) , lm ( g )) . Lemma 3.8.
For f, g, h ∈ ( K [ x ])[ ˜ x ] \ K [ x ] , if lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i ,then we have the following triangular relationship among their S -polynomials: λS ( f, g ) = λ · lcm( lt ( f ) , lt ( g ))lcm( lt ( f ) , lt ( h )) S ( f, h ) − λ · lcm( lt ( f ) , lt ( g ))lcm( lt ( h ) , lt ( g )) S ( g, h ) , (3.9) where the multiplier λ := lc ( h ) /d with d := gcd(lcm( lc ( f ) , lc ( g )) , lc ( h )) ∈ K [ x ] . Henceforth we also call the identity (3.9) the triangular identity of S ( f, g ) with respect to h .Proof. From lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i we can easily deduce that:lcm( lm ( f ) , lm ( g )) ∈ h lcm( lm ( f ) , lm ( h )) i ∩ h lcm( lm ( h ) , lm ( g )) i . In order to corroborate that the multiplier λ = lc ( h ) /d suffices to make thetwo fractions in (3.9) terms in ( K [ x ])[ ˜ x ], we only need to consider the case whenmult p ( lc ( h )) > γ := max { mult p ( lc ( f )) , mult p ( lc ( g )) } for every irreducible factor p of lc ( h ). In this case we have mult p (lcm( lc ( f ) , lc ( h ))) = mult p ( lc ( h )) =mult p (lcm( lc ( h ) , lc ( g ))) in the denominators of (3.9). Hence in the numeratorsof (3.9) we can take mult p ( λ ) = mult p ( lc ( h )) − γ = mult p ( lc ( h )) − mult p ( d ).Now let us write the definition of S -polynomial in (3.1) into the following form: S ( f, g ) = lcm( lt ( f ) , lt ( g )) lt ( f ) f − lcm( lt ( f ) , lt ( g )) lt ( g ) g. (3.10)Then the identity (3.9) readily follows if we also write S ( f, h ) and S ( g, h ) into theform of (3.10). 10t is easy to verify that the identity (3.9) is consistent with the inequality (3.2)for S -polynomials since from (3.9) we can deduce that lm ( S ( f, g )) is dominatedby one of the following leading monomials:lcm( lm ( f ) , lm ( g ))lcm( lm ( f ) , lm ( h )) lm ( S ( f, h )) , or lcm( lm ( f ) , lm ( g ))lcm( lm ( h ) , lm ( g )) lm ( S ( g, h )) . For f ∈ ( K [ x ])[ ˜ x ] \ K [ x ] and g ∈ ( K [ x ]) ∗ , we shall use the following relationbetween (3.1) and (3.3): S ( f, g · lm ( f )) = S ( f, g ) . (3.11)Moreover, the S -polynomial in (3.3) coincides with the term pseudo-reductionin (2.1) for g ∈ R ∗ = ( K [ x ]) ∗ . Algorithm 3.9 (Pseudo-eliminant of a zero-dimensional ideal over a PID) . Input: A finite polynomial set F ⊂ ( K [ x ])[ ˜ x ] \ K .Output: A pseudo-eliminant χ ε ∈ ( K [ x ]) ∗ , pseudo-basis B ε ⊂ h F i \ K [ x ] andmultiplier set Λ ⊂ K [ x ] \ K .Initialization: A temporary basis set G := F \ K [ x ]; a multiplier set Λ := ∅ in K [ x ]; a temporary set S := ∅ in ( K [ x ])[ ˜ x ] \ K for S -polynomials. If F ∩ K [ x ] = ∅ , we initialize f := gcd( F ∩ K [ x ]); otherwise we initialize f := 0.For each pair f, g ∈ G with f = g , we invoke Procedure Q as follows tocompute their S -polynomial S ( f, g ).Procedure Q : Input: f, g ∈ ( K [ x ])[ ˜ x ] \ K [ x ] .If lm ( f ) and lm ( g ) are relatively prime, we define d := gcd( lc ( f ) , lc ( g )) as in (3.5) . If d ∈ K [ x ] \ K , we add d into the multiplier set Λ . Then wedisregard the S -polynomial S ( f, g ) .If there exists an h ∈ G \ { f, g } such that lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i , andthe triangular identity (3.9) has never been applied to the same triplet { f, g, h } ,we compute the multiplier λ as in (3.9) . If λ ∈ K [ x ] \ K , we add λ into themultiplier set Λ . Then we disregard the S -polynomial S ( f, g ) .If neither of the above two cases is true, we compute their S -polynomial S ( f, g ) as in (3.1) . Then we add S ( f, g ) into the set S . End of Q We recursively repeat Procedure P as follows for the pseudo-reductions of allthe S -polynomials in the set S .Procedure P : For an S ∈ S , we invoke Theorem 2.7 to make a pseudo-reduction of S bythe temporary basis set G .If the multiplier λ ∈ K [ x ] \ K in (2.2) , we add λ into the multiplier set Λ .If the remainder r ∈ ( K [ x ])[ ˜ x ] \ K [ x ] , we add r into G . For every f ∈ G \ { r } , we invoke Procedure Q to compute the S -polynomial S ( f, r ) .If the remainder r ∈ K [ x ] \ K , we redefine f := gcd( r, f ) .If the remainder r ∈ K ∗ , we halt the algorithm and output G = { } .Then we delete S from the set S . End of P Finally we define χ ε := f and B ε := G respectively.Procedure R : 11 or every f ∈ B ε , if d := gcd( lc ( f ) , χ ε ) ∈ K [ x ] \ K , we add d into themultiplier set Λ . End of R We output χ ε , B ε and Λ. Remark . In Algorithm 3.9 we compute both d := gcd( lc ( f ) , lc ( g )) when lm ( f ) and lm ( g ) are relatively prime in Procedure Q and d := gcd( lc ( f ) , χ ε ) forevery f ∈ B ε in Procedure R . The purpose of these computations is to procurethe multipliers d in (3.5) of Lemma 3.6 and (3.4) of Lemma 3.5 respectively forthe pseudo-reductions of the S -polynomials. It is the reason why we add d intothe multiplier set Λ when d ∈ K [ x ] \ K .Moreover, in Procedure Q the condition that there exists an h ∈ G \ { f, g } such that lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i is a condition for Lemma 3.8. Definition 3.11 (Pseudo-eliminant; pseudo-basis; multiplier set) . We call the univariate polynomial χ ε obtained via Algorithm 3.9 a pseudo-eliminant of the zero-dimensional ideal I . We also call the polynomial set B ε a pseudo-basis of I and Λ its multiplier set .Please note that contrary to the convention, we do not include the pseudo-eliminant χ ε in the pseudo-basis B ε since we shall contrive modular algorithms tocompute modular bases with the factors of χ ε as moduli in Section 5. Lemma 3.12. (i) A pseudo-eliminant χ ε of a zero-dimensional ideal I is divisibleby its eliminant χ . (ii) For each pair f = g in the union set of pseudo-basis andpseudo-eliminant B ε ∪ { χ ε } , the pseudo-reduction of their S -polynomial S ( f, g ) by B ε yields a remainder r ∈ ( χ ε ) in K [ x ] . In particular, this includes the case when r = 0 . (iii) Algorithm 3.9 terminates in a finite number of steps.Proof. (i) The conclusion readily follows from the fact that χ ε ∈ I ∩ K [ x ] = ( χ ).(ii) According to Procedure P in Algorithm 3.9, if the remainder r of thepseudo-reduction by an intermediate polynomial set G satisfies r ∈ ( K [ x ])[ ˜ x ] \ K [ x ], we add it into G . That is, r eventually becomes an element of the pseudo-basis B ε . It is evident that a pseudo-reduction of r by itself per se leads to thezero remainder. On the other hand, if r ∈ K [ x ] \ K , then as per f := gcd( r, f )in Procedure P of Algorithm 3.9, r is divisible by f and hence by the pseudo-eliminant χ ε , i.e., r ∈ ( χ ε ).(iii) The termination of the algorithm follows from the ring ( K [ x ])[ ˜ x ] = K [ x ]being Noetherian. In fact, whenever the remainder r ∈ ( K [ x ])[ ˜ x ] \ K [ x ] in theProcedure P of the algorithm, we add it to the intermediate polynomial set G .As a result, the monomial ideal h lm ( G ) i is strictly expanded since r is pseudo-reduced with respect to G \ { r } . Hence the ascending chain condition imposed onthe chain of ideals h lm ( G ) i ensures the termination of the algorithm.Based on Lemma 3.12 (i), the following conclusion is immediate: Corollary 3.13.
If a pseudo-eliminant χ ε of a zero-dimensional ideal I in K [ x ] satisfies χ ε ∈ K ∗ , then the reduced Gr¨obner basis of I is { } . In what follows we assume that the ideal I is a proper ideal of K [ x ], that is, I = K [ x ]. Thus let us exclude the trivial case of χ ε ∈ K ∗ hereafter. Please refer to [AL94, P48, Definition 1.8.5] or [CLO15, P93, Definition 4] for a definition. Pseudo-eliminant Divisors and Compatibility
Suppose that K is a perfect field whose characteristic is denoted as char( K ). Recallthat finite fields and fields of characteristic zero are perfect fields. In this sectionwe begin to contrive an algorithm retrieving the eliminant χ of a zero-dimensionalideal I from its pseudo-eliminant χ ε obtained in Algorithm 3.9. We first make afactorization of the pseudo-eliminant χ ε into compatible and incompatible divisors.We prove that the compatible divisors of χ ε are the authentic factors of χ . Thisshows that Algorithm 3.9 generates the eliminant χ when the pseudo-eliminant χ ε is compatible. We compute the factors of χ that correspond to the incompatibledivisors of χ ε in Section 5. Definition 4.1 (Squarefree factorization of univariate polynomials) . A univariate polynomial f ∈ K [ x ] \ K is squarefree if it has no quadraticfactors in K [ x ] \ K . That is, for every irreducible polynomial g ∈ K [ x ] \ K , f isnot divisible by g .The squarefree factorization of a univariate polynomial f ∈ K [ x ] \ K refers toa product f = Q si =1 g ii with g s ∈ K [ x ] \ K such that for those g i ’s that are notconstants, they are both squarefree and pairwise relatively prime. Moreover, the squarefree part of f is defined as Q si =1 g i .The squarefree factorization is unique up to multiplications by constants in K ∗ . Its existence and uniqueness follow from the fact that K [ x ] is a PID andhence a unique factorization integral domain. There are various algorithms forsquarefree factorization depending on the field K being finite or of characteristiczero. Algorithm 4.2 as follows amalgamates these two cases of field characteristics.We improve the algorithm in [GP08, P539, Algorithm B.1.6] over a finite field andthen apply it to the squarefree factorization over a filed of characteristic zero.Consider the integer set J := N ∗ when char( K ) = 0, and J := N \ p N whenchar( K ) = p >
0. That is, J stands for the set of positive integers that are nota multiple of p when char( K ) = p >
0. Let us enumerate the positive integers in J by the bijective enumeration map ρ : N ∗ → J such that ρ ( i ) < ρ ( j ) whenever i < j . We have the evident inequality ρ ( i ) ≥ i when char( K ) = p >
0. Whenchar( K ) = 0, the enumeration map ρ is the identity map. In the algorithm below,for every i ∈ N ∗ we simply denote its image ρ ( i ) ∈ J as [ i ], i.e., ρ ( i ) = [ i ]. Algorithm 4.2 (Squarefree factorization of a univariate polynomial) . Input: A univariate polynomial f ∈ K [ x ] \ K .Output: The squarefree decomposition { g , . . . , g s } of f .Procedure P : If f ′ = 0 , we compute the greatest common divisor f [1] := gcd( f, f ′ ) first.The squarefree part of f is defined as h [1] := f /f [1] .We repeat the following procedure starting with i = 1 until i = s such that f ′ [ s ] = 0 : h [ i +1] := gcd( f [ i ] , h [ i ] ); g [ i ] := h [ i ] /h [ i +1] . (cid:26) f [ i +1] := f [ i ] /h [ i +1] − [ i ][ i +1] f [ i +1] := f [ i ] /h [ i +1] . if char( K ) > if char( K ) = 0 . If char( K ) >
0, the exponent [ i + 1] − [ i ] of h [ i +1] in the following definition of f [ i +1] is theimprovement on [GP08, P539, Algorithm B.1.6]. f f [ s ] ∈ K , we define g [ s ] := h [ s ] to obtain the squarefree factorization f = Q si =1 g [ i ][ i ] .If char( K ) = p > and f [ s ] ∈ K [ x ] \ K , we invoke Procedure Q on f [ s ] . End of P Procedure Q : If char( K ) = p > and f ∈ K [ x ] \ K satisfies f ′ = 0 , we repeat the followingprocedure starting with x := x and ψ := f until i = t such that ψ ′ t = 0 : x i +1 := x pi ; ψ i +1 ( x i +1 ) := ψ i ( x i ) We treat ψ t as f and invoke Procedure P on ψ t . End of Q Procedure Q in Algorithm 4.2 is a composition of Frobenius automorphism. Proposition 4.3.
We can procure a squarefree factorization of f in finite stepsvia Algorithm 4.2.Proof. The termination of the algorithm in finite steps readily follows from the factthat deg f [ i +1] < deg f [ i ] in Procedure P as well as deg ψ i +1 < deg ψ i in Procedure Q . In the case of char( K ) = 0, the bijective map ρ : N ∗ → J is an identity mapsuch that ρ ( i ) = [ i ] = i . To illustrate the procedure of the algorithm, supposethat f = Q si =1 g ii is a squarefree factorization of f . Then h = Q si =1 g i is thesquarefree part of f obtained in the beginning of Procedure P . Moreover, the h i in Procedure P equals Q sj = i g j for 1 ≤ i ≤ s . Hence we have g i = h i /h i +1 . Further, f i equals Q sj = i +1 g j − ij for 1 ≤ i < s . Finally f s = 1 and Procedure P terminatessince f ′ s = 0. Thus follows the squarefree factorization.In the case when char( K ) = p >
0, suppose that f = Q k ∈ p N g kk · Q si =1 g [ i ][ i ] . Letus denote ϕ p := Q k ∈ p N g kk and ϕ q := Q si =1 g [ i ][ i ] such that f = ϕ p ϕ q . Then the f [1] in Procedure P equals ϕ p · Q si =2 g [ i ] − i ] . Hence h [1] = Q si =1 g [ i ] is the squarefree partof ϕ q . And the h [ i ] equals Q sj = i g [ j ] for 1 ≤ i ≤ s . Hence we have g [ i ] = h [ i ] /h [ i +1] .Moreover, f [ i ] equals ϕ p · Q sj = i +1 g [ j ] − [ i ][ j ] for 1 ≤ i < s . Finally, f [ s ] = ϕ p and thus f ′ [ s ] = 0. Now we have a squarefree factorization of ϕ q as Q si =1 g [ i ][ i ] .Procedure Q amounts to a variable substitution x t = x p t such that f [ s ] ( x ) = ψ t ( x t ). Since ψ ′ t = 0, we treat ψ t as f and assume that ψ t = ϕ p ϕ q with ϕ p and ϕ q being defined as above. We repeat Procedure P on ψ t to obtain a squarefreefactorization ϕ q = Q si =1 g [ i ][ i ] . Nonetheless here ϕ q is in the variable x t . Since thefield K is perfect, we have ϕ q ( x t ) = ( ϕ q ( x )) p t = Q si =1 g [ i ] p t [ i ] . Remark . A remarkable thing about Algorithm 4.2 is that as long as the field K is perfect, it is independent of K and any of its field extensions. In fact, allthe computations in Procedure P are based on f [1] = gcd( f, f ′ ) and h [ i +1] :=gcd( f [ i ] , h [ i ] ) that are independent of the field extensions of K .We do not attempt to make a complete factorization of a univariate polynomialdue to the complexity of distinct-degree factorizations. There is a discussion onthe various stages of univariate polynomial factorizations including both squarefreeand distinct-degree factorizations on [GG13, P379].14 efinition 4.5 (Compatible and incompatible divisors and parts) . For a zero-dimensional ideal I over a perfect field K , let χ ε be a pseudo-eliminant of I . Assume that Λ is the multiplier set for the pseudo-reductions ofall the S -polynomials as in Algorithm 3.9. For an irreducible factor p of χ ε withmultiplicity i , if there exists a multiplier λ ∈ Λ such that λ is divisible by p , then p i is called an incompatible divisor of χ ε . If p is relatively prime to every multiplier λ in Λ, then p i is called a compatible divisor of χ ε .We name the product of all the compatible divisors of χ ε as the compatiblepart of χ ε and denote it as cp ( χ ε ). The incompatible part of χ ε is defined as theproduct of all the incompatible divisors of χ ε and denoted as ip ( χ ε ). In particular,we say that a pseudo-eliminant χ ε per se is compatible if it has no incompatibledivisors.From the above Definition 4.5 it is evident that χ ε = cp ( χ ε ) · ip ( χ ε ). Inthe following Algorithm 4.6, we compute the compatible part cp ( χ ε ) and make asquarefree decomposition of the incompatible part ip ( χ ε ) simultaneously. Algorithm 4.6 (Compatible part cp ( χ ε ) of a pseudo-eliminant χ ε and squarefreedecomposition of its incompatible part ip ( χ ε )) . Input: A pseudo-eliminant χ ε ∈ K [ x ] and multiplier set Λ ⊂ K [ x ] that areobtained from Algorithm 3.9.Output: Compatible part cp ( χ ε ) and squarefree decomposition { Ω i : 1 ≤ i ≤ s } of the incompatible part ip ( χ ε ) with Ω i ⊂ K [ x ].We invoke Algorithm 4.2 to make a squarefree factorization χ ε = Q si =1 q ii .For each multiplicity i satisfying 1 ≤ i ≤ s , we construct a polynomial setΩ i ⊂ K [ x ] whose elements are pairwise relatively prime as follows:For every λ ∈ Λ, we compute d λi := gcd( λ, q i ). If d λi ∈ K [ x ] \ K , we checkwhether d λi is relatively prime to every element ω that is already in Ω i . If not,we substitute d λi by d λi / gcd( d λi , ω ). And we substitute the ω in Ω i by bothgcd( d λi , ω ) and ω/ gcd( d λi , ω ). We repeat the process until either d λi ∈ K , or d λi is relatively prime to every element in Ω i . We add d λi into Ω i if d λi ∈ K [ x ] \ K .Finally, for each multiplicity i satisfying 1 ≤ i ≤ s , we compute the product ω i := Q ω ∈ Ω i ω . Then we output χ ε / Q si =1 ω ii as the compatible part cp ( χ ε ). Wealso output { Ω i : 1 ≤ i ≤ s } as a squarefree decomposition of the incompatiblepart ip ( χ ε ). Definition 4.7 (Composite divisor set Ω i ; composite divisor ω i ) . We call the univariate polynomial set Ω i for 1 ≤ i ≤ s obtained in Algorithm4.6 a composite divisor set of the incompatible part ip ( χ ε ) of the pseudo-eliminant χ ε . For an element ω of Ω i , we call its i -th power ω i a composite divisor of theincompatible part ip ( χ ε ).A composite divisor ω i is a product of the incompatible divisors p i in Definition4.5. The incompatible part ip ( χ ε ) is the product of all the composite divisors ω i according to the final step of Algorithm 4.6, that is: ip ( χ ε ) = s Y i =1 Y ω ∈ Ω i ω i . (4.1)The above composite divisors ω i are pairwise relatively prime by the constructionof the composite divisor set Ω i in Algorithm 4.6.15t is evident that Algorithm 4.6 terminates in finite steps since the multiplierset Λ in Algorithm 4.6 is a finite set. Lemma 4.8.
Suppose that F = { f j : 1 ≤ j ≤ s } ⊂ ( K [ x ])[ ˜ x ] \ K [ x ] is apolynomial set. Moreover, each f j has the same leading monomial lm ( f j ) = ˜ x α ∈ [ ˜ x ] for ≤ j ≤ s .(i) If f = P sj =1 f j satisfies lm ( f ) ≺ ˜ x α , then there exist multipliers b, b j ∈ ( K [ x ]) ∗ for ≤ j < s such that bf = X ≤ j
Suppose that χ is the eliminant of a zero-dimensional ideal I in ( K [ x ])[ ˜ x ] over a perfect field K and χ ε a pseudo-eliminant of I . Then χ isdivisible by the compatible divisors of χ ε , that is, for every compatible divisor p i of χ ε , we have mult p ( χ ε ) = mult p ( χ ) = i . Hence χ is divisible by the compatiblepart cp ( χ ε ) of χ ε . In particular, χ = χ ε if χ ε per se is compatible.Proof. With p ∈ K [ x ] \ K being an irreducible polynomial and i ∈ N ∗ , let p i be acompatible divisor of the pseudo-eliminant χ ε as in Definition 4.5. We shall provethat the eliminant χ is also divisible by p i . That is, mult p ( cp ( χ ε )) = i ≤ mult p ( χ ).Thus as per Lemma 3.12 (i) we have:mult p ( χ ε ) = mult p ( cp ( χ ε )) = mult p ( χ ) = i. (4.7)Let G ∪ { f } := { f j : 0 ≤ j ≤ s } ⊂ ( K [ x ])[ ˜ x ] \ K be the basis of the ideal I after the Initialization in Algorithm 3.9 with f ∈ K [ x ] \ K ∗ . In the genericcase when f = 0, i.e., f ∈ K [ x ] \ K , the definition of pseudo-eliminant χ ε inAlgorithm 3.9 shows that f ∈ ( χ ε ) ⊂ K [ x ]. That is, there exists ρ ∈ ( K [ x ]) ∗ such that f = ρχ ε . The eliminant χ ∈ I ∩ K [ x ] can be written as: χ = s X j =0 h j f j = s X j =1 h j f j + ρh χ ε (4.8)with h j ∈ ( K [ x ])[ ˜ x ] for 0 ≤ j ≤ s . Let us abuse the notation a bit and denote F := G ∪ { f } = { f j : 0 ≤ j ≤ s } . Suppose that max ≤ j ≤ s { lm ( h j f j ) } = ˜ x β . Letus collect and rename the elements in the set { f j ∈ F : lm ( h j f j ) = ˜ x β , ≤ j ≤ s } into a new set B t := { g j : 1 ≤ j ≤ t } . And the subscripts of the functions { h j } are adjusted accordingly. In this way (4.8) can be written as follows: χ = t X j =1 h j g j + X f j ∈ F \ B t h j f j , (4.9)where the products h j f j are those in (4.8) with f j ∈ F \ B t for 0 ≤ j ≤ s .If we denote lt ( h j ) := c j ˜ x α j with c j ∈ ( K [ x ]) ∗ for 1 ≤ j ≤ t in (4.9), then itis evident that the following polynomial: g := t X j =1 lt ( h j ) · g j = t X j =1 c j ˜ x α j g j (4.10)is a summand of (4.9) and satisfies lm ( g ) ≺ ˜ x β = lm ( ˜ x α j g j ) for 1 ≤ j ≤ t sincethe eliminant χ ≺ ˜ x β in (4.9). According to Lemma 4.8 (i), there exist multipliers b, b j ∈ ( K [ x ]) ∗ for 1 ≤ j < t that satisfy the following identity: bg = X ≤ j For a zero-dimensional ideal I over a perfect field K , let χ ε bea pseudo-eliminant of I and B ε its pseudo-basis. Suppose that p is an irreduciblefactor of χ ε with multiplicity i . If p is relatively prime to every leading coefficientin lc ( B ε ) := { lc ( b ) ∈ K [ x ] \ K : b ∈ B ε } , then p i is a compatible divisor of χ ε .In particular, if χ ε is relatively prime to every leading coefficient in lc ( B ε ) , then χ ε is compatible and χ = χ ε .Proof. Let us prove that the divisor p i of χ ε satisfies Definition 4.5 and henceis compatible when p is relatively prime to every leading coefficient in lc ( B ε ).That is, p i is relatively prime to the multiplier set Λ for the pseudo-reductionsof S -polynomials by the pseudo-basis B ε in obtaining χ ε . In what follows let uswrite an S -polynomial as S and the pseudo-basis as B ε = { b , . . . , b s } . Thenthe multiplier λ ∈ ( K [ x ]) ∗ in (2.2) for the pseudo-reduction of S by B ε is justa finite product of the interim multipliers µ := m/c α ∈ ( K [ x ]) ∗ in (2.1) with m = lcm( c α , lc ( b j )). It is evident that if an irreducible factor p of χ ε is relativelyprime to lc ( b j ), then it is also relatively prime to the interim multiplier µ = m/c α since m/c α = lc ( b j ) / gcd( c α , lc ( b j )). As a result, p is relatively prime to themultiplier λ for the pseudo-reduction of S by B ε . Thus p i is a compatible divisorof χ ε .Definition 4.5 provides a criterion for the compatibility of a pseudo-eliminant’sdivisors based on the multiplier set Λ, whereas the criterion in Corollary 4.11 isbased on the leading coefficients of the pseudo-basis B ε . To distinguish we callthem the multiplier criterion and coefficient criterion respectively. The followingexample shows that a divisor of a pseudo-eliminant χ ε satisfying the multipliercriterion does not necessarily satisfy the coefficient criterion. Example . In the algebra ( Q [ y ])[ x ] with elimination ordering x ≻ y , consideran ideal I generated by f ( x, y ) = y ( x + 1) and g ( x, y ) = ( y + 1)(2 x + 1). As per(3.1), their S -polynomial is as follows with lc ( f ) = y and lc ( g ) = 2( y + 1). S ( f, g ) = 2( y + 1) f − yxg = − y ( y + 1)( x − . The pseudo-reduction of S ( f, g ) is 2 S ( f, g ) + yg = 5 y ( y + 1) with the multiplier λ = 2 ∈ Q . The pseudo-eliminant χ ε = 5 y ( y + 1) and pseudo-basis F = { f, g } .Now both the irreducible factors y and y + 1 of χ ε satisfy the multiplier criterionand hence are compatible divisors. Hence χ ε = 5 y ( y + 1) is compatible and theeliminant χ = χ ε up to the unit coefficient 5 ∈ Q . Nevertheless χ ε does not satisfythe coefficient criterion. For a pseudo-eliminant χ ε of a zero-dimensional ideal I in ( K [ x ])[ ˜ x ] over a perfectfield K , we made a squarefree decomposition of its incompatible part ip ( χ ε ) inAlgorithm 4.6. In order to determine the eliminant χ of I , we perform a complete21nalysis of ip ( χ ε ) in this section. For the squarefree decomposition { Ω i } of ip ( χ ε )obtained in Algorithm 4.6, the elements in Ω i are pairwise relatively prime andusually have small exponents due to the way they are constructed in Algorithm4.6. Accordingly it is natural to contrive a modular algorithm modulo the elementsin Ω i so as to reduce the complexity of our algorithm. However this requiresunorthodox computations in principal ideal rings with zero divisors. Definition 5.1 (Principal ideal Quotient Ring : PQR) . Let R be a PID whose set of units is denoted as R × . With the principal ideal( q ) generated by an element q ∈ R ∗ \ R × , the quotient ring ¯ R := R/ ( q ) is calleda principal ideal quotient ring and abbreviated as PQR henceforth. Notation . Suppose that q ∈ R ∗ \ R × in Definition 5.1 has a unique factorization q = u Q si =1 p α i i with s, α i ∈ N ∗ for 1 ≤ i ≤ s and u ∈ R × . Here the factors { p i : 1 ≤ i ≤ s } ⊂ R ∗ \ R × are irreducible and they are pairwise relatively primewhen s > q ∈ R ∗ \ R × is irreducible in Definition 5.1, i.e., when s = α = 1 inNotation 5.2, the PQR ¯ R becomes a field since q is prime in R . Nonetheless when q ∈ R ∗ \ R × is not irreducible in Definition 5.1, i.e., in the case of either s > α i > i satisfying 1 ≤ i ≤ s , the PQR ¯ R has zero divisors and is not anintegral domain. In this case ¯ R is no longer a factorial ring. Nonetheless ¯ R stillhas nice properties to which we can resort in our computations. Lemma 5.3. Let ¯ R = R/ ( q ) be a PQR as in Definition 5.1 and ϕ : R → ¯ R thecanonical ring homomorphism. Suppose that q ∈ R ∗ \ R × has a unique factorization q = u Q si =1 p α i i as in Notation 5.2. In what follows we also use the notation ¯ a := ϕ ( a ) for every a ∈ R .(i) An r ∈ R is relatively prime to q , i.e., gcd( r, q ) = 1 , if and only if ϕ ( r ) is aunit in ¯ R , that is, ϕ ( r ) ∈ ¯ R × .(ii) For ≤ i ≤ s and each l ∈ N satisfying l ≥ α i , we have ¯ p li ∼ ¯ p α i i . Here thenotation ¯ a ∼ ¯ b in ¯ R means that ¯ a is an associate of ¯ b in ¯ R , i.e., there is aunit ¯ u ∈ ¯ R × such that ¯ b = ¯ u ¯ a .(iii) For every ¯ a ∈ ¯ R ∗ , we have a unique representation ¯ a ∼ Q si =1 ¯ p β i i that satisfies ≤ β i ≤ α i for ≤ i ≤ s . We call such kind of representations a standard representation of ¯ a in the PQR ¯ R and denote it as ¯ a st . In particular, wedefine ¯ a st := 1 for ¯ a ∈ ¯ R × .Proof. The conclusion (i) readily follows from the fact that R is a PID. In fact,gcd( r, q ) = 1 means that there exist u, v ∈ R such that ur + vq = 1, from whichwe can deduce that ϕ ( u ) ϕ ( r ) = 1. Conversely ϕ ( ur ) = 1 implies that there exists v ∈ R such that ur − vq . Hence follows the conclusion.When s = 1 both the conclusions (ii) and (iii) are evident since ¯ p l = 0 for l ≥ α . In particular, ¯ R ∗ = ¯ R × when α = s = 1. In this case every ¯ a ∈ ¯ R ∗ hasa standard representation ¯ a ∼ a st . So in what follows let us suppose that s > i = s . For l > α s , we have p ls ≡ p ls + q mod q . Moreover, we have the identity: p ls + q =22 α s s ( p l − α s s + u Q s − i =1 p α i i ). Here ϕ ( p l − α s s + u Q s − i =1 p α i i ) is a unit in ¯ R by (i) since p l − α s s + u Q s − i =1 p α i i is relatively prime to q in R . Thus ¯ p ls = ϕ ( p ls + q ) ∼ ¯ p α s s .The existence of the standard representation in (iii) readily follows from thefact that R is factorial and the canonical homomorphism ϕ is an epimorphism. If p is irreducible and relatively prime to q , then ¯ p = ϕ ( p ) ∈ ¯ R × is a unit. Hencefor every ¯ a ∈ ¯ R ∗ , its standard representation can only bear the form ¯ a ∼ Q si =1 ¯ p β i i with 0 ≤ β i ≤ α i for 1 ≤ i ≤ s . Now suppose that ¯ a has another standardrepresentation ¯ a ∼ Q si =1 ¯ p γ i i with 0 ≤ γ i ≤ α i for 1 ≤ i ≤ s . Then there exists h ∈ R such that Q si =1 p β i i = u · Q si =1 p γ i i + hq with u ∈ R being relatively prime to q , from which we can easily deduce that β i = γ i for 1 ≤ i ≤ s .Let K be a perfect field and q ∈ K [ x ] \ K . It is easy to see that K [ x ] / ( q )is a PQR as defined in Definition 5.1. Hereafter we use R and ¯ R to denote K [ x ]and K [ x ] / ( q ) respectively. Let us consider the following set: R q := { r ∈ K [ x ] : deg( r ) < deg( q ) } (5.1)with deg( r ) = 0 for every r ∈ K including r = 0. Let us deem the canonical ringhomomorphism ϕ : R → ¯ R as a map. We restrict it on R q and denote it as ϕ q . Itis evident that ϕ q : R q → ¯ R is a bijective map with ϕ q (0) = 0. We redefine thetwo binary operations on R q , the addition and multiplication, as follows. a + b := ϕ − q (¯ a + ¯ b ); a · b := ϕ − q (¯ a · ¯ b ) . (5.2)In this way the set R q in (5.1) becomes a ring, which we still denote as R q . It iseasy to verify that ϕ q is a ring isomorphism between R q and ¯ R . As a result, theconclusions in Lemma 5.3 apply to the ring R q as well. Definition 5.4 (Normal PQR R q ) . We call the ring R q being constructed as in (5.1) and (5.2) a normal PQRhenceforth.Let a normal PQR R q be defined as in Definition 5.4 for q ∈ K [ x ] \ K . Forevery f ∈ R = K [ x ], there exist a quotient h ∈ K [ x ] and unique remainder r ∈ K [ x ] satisfying f = hq + r ; deg( r ) < deg( q ) . (5.3)Hence by (5.1) we can define an epimorphism directly as follows. σ q : R → R q : σ q ( f ) := r. (5.4)It is easy to verify that the epimorphism σ q is a composition of the canonicalring homomorphism ϕ : R → ¯ R = K [ x ] / ( q ) and the isomorphism ϕ − q : ¯ R → R q in (5.2).Since a normal PQR R q is a subset of R = K [ x ], for every r ∈ R q , we candefine an injection as follows. ι q : R q ֒ → R : ι q ( r ) := r. (5.5)Please note that ι q is not a ring homomorphism since the binary operations onthe ring R q are different from those on R . Nonetheless σ q ◦ ι q is the identity map23n R q . For each pair a, b ∈ R q , we define the binary operations between ι q ( a ) and ι q ( b ) as those defined on R .Suppose that ¯ a ∈ ¯ R ∗ has a standard representation ¯ a ∼ ¯ a st = Q si =1 ¯ p β i i with0 ≤ β i ≤ α i as in Lemma 5.3 (iii). We can substitute p i = ϕ − q (¯ p i ) ∈ R ∗ q \ R × q asin (5.2) for ¯ p i ∈ ¯ R ∗ \ ¯ R × in this representation. In this way we obtain a standard representation of a := ϕ − q (¯ a ) ∈ R ∗ q in the normal PQR R q as follows: a ∼ a st := s Y i =1 p β i i , ≤ β i ≤ α i ; a = a × · a st , (5.6)where { α i : 1 ≤ i ≤ s } are the exponents for the unique factorization of the moduli q as in Lemma 5.3. For convenience we use a × ∈ R × q to denote the unit factorof a with respect to a st . We also call a st the standard factor of a henceforth.In particular, we define a st := 1 for a ∈ R × q . We can derive the existence anduniqueness of the standard representation a st in (5.6) from Lemma 5.3 (iii) sincethe normal PQR R q is isomorphic to the PQR ¯ R under ϕ q . Remark . It is unnecessary to procure a complete factorization of a st as in (5.6)in our computations. In fact, it suffices to make a factorization a = a × · a st . Thiscan be easily attained by a computation a st = gcd( ι q ( a ) , q ) with ι q being definedas in (5.5). The soundness of the computation readily follows from Lemma 5.3and (5.6).An apparent difference between the PQR ¯ R and normal PQR R q is that thedegree function deg is well defined on R q , which is indispensable for polynomialdivisions. More specifically, for all a, b ∈ R ∗ q with deg( b ) > 0, there exist a quotient h ∈ R q and unique remainder r ∈ R q satisfying the following equality: a = hb + r such that deg( r ) < deg( b ) . (5.7)Since all polynomials involved here including the product hb have degrees strictlyless than deg( q ) in (5.7), there is no multiplication of zero divisors leading to 0for polynomial divisions. This includes the case when deg( a ) < deg( b ) and hence h = 0. That is, we make polynomial divisions on the normal PQR R q in the sameway as on R .For a, b ∈ R ∗ q and their standard representations a ∼ a st = Q si =1 p β i i and b ∼ b st = Q si =1 p γ i i as in (5.6), let us define:gcd st ( a, b ) := gcd( a st , b st ) = s Y i =1 p n i i ; lcm st ( a, b ) := lcm( a st , b st ) = s Y i =1 p m i i (5.8)with n i := min { β i , γ i } and m i := max { β i , γ i } . It is evident that we might havelcm st ( a, b ) = 0 for a, b ∈ R ∗ q due to the possible existence of zero divisors in R ∗ q . Remark . The definition of gcd st ( a, b ) and lcm st ( a, b ) in (5.8) is based upon acomplete factorization of a and b as in (5.6). In practice in order to minimizethe complexity of our algorithm, we resort to Euclidean algorithm to computegcd( a, b ). The normal PQR R q might have zero divisors and not be an Euclideandomain. However from our discussion on polynomial divisions in (5.7), we knowthat the polynomial division on R q is the same as that on R . Moreover, for the24rreducible factor p i ∈ R ∗ q \ R × q in Notation 5.2 and 1 ≤ e ≤ α i , if both a and b are divisible by p ei in (5.7), then so is the remainder r . Similarly if both b and r are divisible by p ei in (5.7), then so is a . Thus the computation of gcd( a, b )for a, b ∈ R ∗ q by Euclidean algorithm on R q is sound and feasible. It differs fromgcd st ( a, b ) only by a unit factor.Let lcm( ι q ( a ) , ι q ( b )) be the least common multiple of ι q ( a ) and ι q ( b ) on R . Thesame for gcd( ι q ( a ) , ι q ( b )). For each pair a, b ∈ R q , let us define:gcd q ( a, b ) := σ q (gcd( ι q ( a ) , ι q ( b ))); lcm q ( a, b ) := σ q (lcm( ι q ( a ) , ι q ( b ))) (5.9)with the epimorphism σ q and injection ι q defined as in (5.4) and (5.5) respectively.By Lemma 5.3 and (5.6), it is easy to verify the following relationship betweenthe two definitions in (5.8) and (5.9):gcd q ( a, b ) ∼ gcd st ( a, b ) and lcm q ( a, b ) ∼ lcm st ( a, b ) . (5.10)In the identity (5.3), mult p ( r ) is well-defined since K [ x ] is a factorial domainand r ∈ K [ x ]. Therefore we can deduce that for every irreducible polynomial p ∈ K [ x ], if max { mult p ( f ) , mult p ( r ) } ≤ mult p ( q ), then we have:mult p ( f ) = mult p ( r ) . (5.11) Definition 5.7 (Elimination ordering on R q [ ˜ x ]) . If the variable x ∈ R ∗ q , the elimination ordering on R q [ ˜ x ] is the monomialordering such that the ˜ x variables are always larger than the variable x ∈ R ∗ q .That is, x α ˜ x γ ≻ x β ˜ x δ if and only if ˜ x γ ≻ ˜ x δ or, ˜ x γ = ˜ x δ and α > β .We also say that the elimination ordering on R q [ ˜ x ] is induced from the one on( K [ x ])[ ˜ x ] in Definition 3.1. Definition 5.8 (Term reduction in R q [ ˜ x ]) . Let R q be a normal PQR as in Definition 5.4 and ≻ the elimination orderingon R q [ ˜ x ] as in Definition 5.7. Let the epimorphism σ q : R → R q and injection ι q : R q → R be defined as in (5.4) and (5.5) respectively. For f ∈ R q [ ˜ x ] \ R q and g ∈ ( R q [ ˜ x ]) ∗ \ R × q with lc ( g ) ∈ R ∗ q , suppose that f has a term c α ˜ x α with˜ x α ∈ supp( f ) ∩ h lm ( g ) i . We also define the multipliers µ := σ q (lcm( l α , l g ) /l α )and m := σ q (lcm( l α , l g ) /l g ) with l α := ι q ( c α ) and l g := ι q ( lc ( g )). We can make a reduction of the term c α ˜ x α of f by g as follows. h = µf − m ˜ x α lm ( g ) g. (5.12)We call h the remainder of the reduction and µ the interim multiplier on f withrespect to g .In Definition 5.8 we might have lcm st ( c α , lc ( g )) = 0 for c α , lc ( g ) ∈ R ∗ q due tothe possible existence of zero divisors in R ∗ q . We postpone to address this issueuntil Lemma 5.17 (ii) after the definition of S -polynomials over a normal PQRbecause in what follows we only consider a special kind of term reductions whoseinterim multipliers µ in (5.12) satisfy µ ∈ R × q .25 efinition 5.9 (Properly reduced polynomial in R q [ ˜ x ]) . Let R q be a normal PQR as in Definition 5.4 and ≻ the elimination orderingon R q [ ˜ x ] as in Definition 5.7. A term c α ˜ x α ∈ R q [ ˜ x ] with the coefficient c α ∈ R ∗ q is said to be properly reducible with respect to F = { f , . . . , f s } ⊂ R q [ ˜ x ] \ R q ifthere exists an f j ∈ F such that ˜ x α ∈ h lm ( f j ) i and the interim multiplier µ withrespect to f j as in (5.12) satisfies µ ∈ R × q . We say that a polynomial f ∈ R q [ ˜ x ] is properly reduced with respect to F if none of its terms is properly reducible withrespect to F .The condition µ ∈ R × q for the properness in Definition 5.9 indicates that µ = σ q (lcm( l α , l j ) /l α ) ∈ R × q . Here l α := ι q ( c α ) and l j := ι q ( c j ) with c j := lc ( f j ). Hencewe can deduce that c α ∈ ( c j ) ⊂ R q . Combined with the condition ˜ x α ∈ h lm ( f j ) i ,the condition µ ∈ R × q for the properness in Definition 5.9 is equivalent to thefollowing term divisibility condition: c α ˜ x α ∈ h c j · lm ( f j ) i = h lt ( f j ) i ⊂ R q [ ˜ x ] . (5.13) Theorem 5.10 (Proper division in R q [ ˜ x ]) . Let R q be a normal PQR as in Definition 5.4 and ≻ the elimination orderingon R q [ ˜ x ] as in Definition 5.7. Suppose that F = { f , . . . , f s } are polynomials in R q [ ˜ x ] \ R q . For every f ∈ R q [ ˜ x ] , there exist a multiplier λ ∈ R × q as well as aremainder r ∈ R q [ ˜ x ] and quotients q j ∈ R q [ ˜ x ] for ≤ j ≤ s such that: λf = s X j =1 q j f j + r, (5.14) where r is properly reduced with respect to F . Moreover, the polynomials in (5.14) have to satisfy the following condition: lm ( f ) = max { max ≤ j ≤ s { lm ( q j ) · lm ( f j ) } , lm ( r ) } . (5.15) Proof. The proof amounts to a verbatim repetition of that for Theorem 2.7 ifwe substitute the criterion of being properly reduced for that of being pseudo-reduced. In fact, the polynomial division on a normal PQR R q as in (5.7) is thesame as that on R = K [ x ]. Moreover, a normal PQR R q as in Definition 5.4 isalso a Noetherian ring since it is isomorphic to the PQR ¯ R in Definition 5.1 thatis Noetherian.Please note that the product lm ( q j ) · lm ( f j ) in (5.15) is based upon the termdivisibility condition (5.13).We also call the proper division in Theorem 5.10 a proper reduction henceforth.We can easily contrive a proper division algorithm based on Theorem 5.10.For f, g ∈ ( R q [ ˜ x ]) ∗ \ R × q , suppose that lcm st ( lc ( f ) , lc ( g )) = 0 due to theexistence of zero divisors in R ∗ q . In this case if we employed Definition 3.4 for S -polynomials directly and in particular, the multiplier m = lcm q ( lc ( f ) , lc ( g )),then their S -polynomial S ( f, g ) would equal 0 since m = lcm q ( lc ( f ) , lc ( g )) = 0in (3.1) as per (5.10). Hence we need to revise Definition 3.4 as follows. Definition 5.11 ( S -polynomial over a normal PQR R q ) . R q be a normal PQR as in Defintion 5.4. Suppose that f ∈ R q [ ˜ x ] \ R q and g ∈ ( R q [ ˜ x ]) ∗ \ R × q . Let us use c f and c g to denote lc ( f ) and lc ( g ) in R ∗ q respectively. With the epimorphism σ q : R → R q and injection ι q : R q → R definedas in (5.4) and (5.5) respectively, we denote l f := ι q ( c f ) and l g := ι q ( c g ). We alsodefine multipliers m f := σ q (lcm( l f , l g ) /l f ) and m g := σ q (lcm( l f , l g ) /l g ) as well asthe monomial ˜ x γ := lcm( lm ( f ) , lm ( g )) ∈ [ ˜ x ] . Then the following polynomial: S ( f, g ) := m f ˜ x γ lm ( f ) f − m g ˜ x γ lm ( g ) g (5.16)is called the S -polynomial of f and g in R q [ ˜ x ].In particular, when f ∈ R q [ ˜ x ] \ R q and g ∈ R ∗ q \ R × q , we can take lm ( g ) = 1and c g = lc ( g ) = g in Definition 5.11. If we define l g := ι q ( g ), the definitions for m f and m g in (5.16) are unaltered. Now ˜ x γ = lm ( f ) and the S -polynomial in(5.16) becomes: S ( f, g ) := m f f − m g g · lm ( f ) . (5.17) Lemma 5.12. For f ∈ R q [ ˜ x ] \ R q and g ∈ R ∗ q \ R × q , and with the same notationsas in (5.17) , let us further denote d := gcd( l f , l g ) . Then the S -polynomial S ( f, g ) in (5.17) satisfies the following identity: S ( f, g ) = σ q (cid:16) l g d (cid:17) ( f − lt ( f )) = m f ( f − lt ( f )) (5.18) with m f = σ q (lcm( l f , l g ) /l f ) being defined as in (5.16) .Proof. It is evident that m f = σ q ( l g /d ) and m g = σ q ( l f /d ). Hence from (5.17)follows directly: S ( f, g ) = σ q (cid:16) l g d (cid:17) f − σ q (cid:16) l f d (cid:17) σ q ( l g ) · lm ( f ) = σ q (cid:16) l g d (cid:17) ( f − lt ( f ))since σ q ◦ ι q is the identity map on R q .There is a special kind of S -polynomials for f ∈ R q [ ˜ x ] \ R q when c f = lc ( f ) ∈ R ∗ q \ R × q . S ( f, q ) := n f f = n f ( f − lt ( f )) (5.19)with n f := σ q (lcm( l f , q ) /l f ) = σ q ( q/ gcd( l f , q )). Here l f := ι q ( c f ) as in (5.16). Lemma 5.13. For f, g ∈ R q [ ˜ x ] \ R q , suppose that lm ( f ) and lm ( g ) are relativelyprime. With the same notations as in Definition 5.11, let us also denote d :=gcd( l f , l g ) . Then their S -polynomial in (5.16) satisfies: S ( f, g ) = f · lt ( g ) − g · lt ( f ) σ q ( d ) = f g − g f gcd q ( lc ( f ) , lc ( g )) (5.20) with f := f − lt ( f ) and g := g − lt ( g ) . Moreover, we have: max { lm ( f ) · lm ( g ) , lm ( g ) · lm ( f ) } ≺ lm ( f ) · lm ( g ) . (5.21)27 roof. In the definition (5.16) we have ˜ x γ = lm ( f ) · lm ( g ). Furthermore, m f = σ q ( l g /d ) and m g = σ q ( l f /d ). Thus the first equality in (5.20) follows from (5.16)as follows. S ( f, g ) = σ q (cid:16) l g d (cid:17) · lm ( g )( f + lt ( f )) − σ q (cid:16) l f d (cid:17) · lm ( f )( g + lt ( g )) , where we can write σ q ( l g /d ) as σ q ( l g ) /σ q ( d ) = c g /σ q ( d ) and same for σ q ( l f /d ).The second equality in (5.20) follows from the first one by substituting g − g and f − f for lt ( g ) and lt ( f ) respectively. And the denominator σ q ( d ) =gcd q ( lc ( f ) , lc ( g )) is defined as in (5.9).The inequality (5.21) is evident since lm ( f ) ≺ lm ( f ) and lm ( g ) ≺ lm ( g ).From Lemma 5.12 and Lemma 5.13 and based on Theorem 5.10, we can easilydeduce the following corollary for algorithmic simplifications. Corollary 5.14. For f, g ∈ R q [ ˜ x ] \ R q , suppose that lm ( f ) and lm ( g ) are relativelyprime. With the same notations as in Lemma 5.13, if σ q ( d ) ∈ R × q , then their S -polynomial S ( f, g ) as in (5.20) can be properly reduced to by f and g as inTheorem 5.10 with the multiplier σ q ( d ) and quotients f and − g .In particular, for f ∈ R q [ ˜ x ] \ R q and g ∈ R ∗ q \ R × q , with the same notations asin Lemma 5.12, if σ q ( d ) = gcd q ( lc ( f ) , g ) ∈ R × q , then their S -polynomial S ( f, g ) as in (5.18) can be properly reduced to by g with the multiplier σ q ( d ) ∈ R × q andquotient f − lt ( f ) . Definition 5.15 ( lcm representation ) . For F = { f , . . . , f s } ⊂ ( R q [ ˜ x ]) ∗ \ R × q , we say that an S -polynomial S ( f, g )has an lcm representation with respect to F if there exist { h , . . . , h s } ⊂ R q [ ˜ x ]satisfying: S ( f, g ) = s X j =1 h j f j such that the following condition holds:max ≤ j ≤ s { lm ( h j ) · lm ( f j ) } ≺ lcm( lm ( f ) , lm ( g )) . (5.22) Remark . The lcm representation is especially suitable for the representationof S -polynomials over such rings with zero divisors as the PQR R q . In particular,the condition (5.21) in Lemma 5.13 amounts to the condition (5.22) for the lcm representation with respect to { f, g } when the multiplier σ q ( d ) ∈ R × q in (5.20) asin Corollary 5.14. Similarly the identity (5.18) is also an lcm representation of S ( f, g ) with respect to g ∈ R ∗ q \ R × q when the multiplier σ q ( d ) ∈ R × q .For g ∈ R ∗ q \ R × q and f ∈ R q [ ˜ x ] \ R q , we shall also use the following relationbetween (5.16) and (5.17): S ( f, g · lm ( f )) = S ( f, g ) . (5.23) Lemma 5.17. (i) The S -polynomial in (5.16) satisfies lm ( S ( f, g )) ≺ ˜ x γ . The S -polynomials in (5.17) and (5.19) satisfy lm ( S ( f, g )) ≺ lm ( f ) and lm ( S ( f, q )) ≺ lm ( f ) respectively. (ii) The two multipliers m f and m g in (5.16) and (5.17) arenot zero, that is, we always have m f , m g ∈ R ∗ q even if lcm q ( lc ( f ) , lc ( g )) = 0 with lcm q defined as in (5.9) . Please refer to [CLO15, P107, Definition 5] for its definition over a field instead. roof. To prove the conclusion (i), it suffices to prove that m f c f = m g c g . Inparticular, we have c g = g in the case of (5.17). The composition σ q ◦ ι q of theepimorphism σ q in (5.4) and injection ι q in (5.5) is the identity map on R q . Hence c f = σ q ( ι q ( c f )) = σ q ( l f ) and we have the following verification: m f c f = σ q (cid:16) lcm( l f , l g ) l f (cid:17) · σ q ( l f ) = σ q (lcm( l f , l g )) . Similarly we have m g c g = σ q (lcm( l f , l g )) and thus the conclusion follows. Theconclusion for (5.19) is easy to corroborate.The conclusion (ii) follows from the identity: m f = σ q (cid:16) lcm( l f , l g ) l f (cid:17) = σ q (cid:16) l g gcd( l f , l g ) (cid:17) . (5.24)In fact, according to the definition of leading coefficients in Notation 2.4, c g = lc ( g ) ∈ R ∗ q . Hence l g = ι q ( c g ) ∈ ( K [ x ]) ∗ and deg( l g ) < deg( q ) as in (5.1). Thusthe multiplier m f ∈ R ∗ q by (5.24). The same holds for m g .A conspicuous difference between the S -polynomials in Definition 5.11 over anormal PQR and those in Definition 3.4 over a PID is that the leading coefficients m f · lc ( f ) = m f c f and m g · lc ( g ) = m g c g in (5.16) and (5.17) might be zero due tothe possible existence of zero divisors in R ∗ q . We shall prove that this imposes nohindrance to the viability of our computations. For S -polynomials over a normalPQR R q , Lemma 5.17 (i) is equivalent to the inequality (3.2).For f, g ∈ ( R q [ ˜ x ]) ∗ \ R × q , let us define:lcm q ( lt ( f ) , lt ( g )) := lcm q ( lc ( f ) , lc ( g )) · lcm( lm ( f ) , lm ( g )) (5.25)with lcm q being defined as in (5.9).Let us use the same notations as in Definition 5.11 for S -polynomials. For f, g ∈ ( R q [ ˜ x ]) ∗ \ R × q without both of them in R ∗ q \ R × q , we define the lcm multiplier of f and g as:cmr( g | f ) := σ q (cid:16) lcm( l f , l g ) l f (cid:17) lcm( lm ( f ) , lm ( g )) lm ( f ) = m f ˜ x γ lm ( f ) . (5.26)Then the definition for the S -polynomial S ( f, g ) in (5.16) can be written as: S ( f, g ) = cmr( g | f ) · f − cmr( f | g ) · g. (5.27) Lemma 5.18. For f, g, h ∈ ( R q [ ˜ x ]) ∗ \ R × q with at most one of them in R ∗ q \ R × q ,if lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i , then we have the following relationship betweentheir S -polynomials: λS ( f, g ) = λ · cmr( g | f )cmr( h | f ) S ( f, h ) − λ · cmr( f | g )cmr( h | g ) S ( g, h ) (5.28) with the lcm multiplier cmr being defined as in (5.26) . Here the multiplier λ := σ q ( l h /d ) ∈ R ∗ q with l h := ι q ( lc ( h )) and d := gcd(lcm( l f , l g ) , l h ) . roof. Same as the conclusion in Lemma 5.17 (ii), the denominators cmr( h | f ) andcmr( h | g ) in (5.28) are nonzero, which is the reason why we use the lcm multipliercmr as in (5.26) instead of the lcm q as in (5.25).The multiplier λ := σ q ( l h /d ) ∈ R ∗ q can indeed render the two fractions in(5.28) terms in R q [ ˜ x ]. The proof is totally similar to that for the multiplier λ inthe identity (3.9) in Lemma 3.8.We can corroborate the identity in (5.28) directly by the form of S -polynomialsin (5.27) as well as the definition of lcm multipliers in (5.26).Based on the above discussions we now analyze the incompatible part ip ( χ ε )of the pseudo-eliminant χ ε . Our goal is to determine the corresponding factors ofthe eliminant χ of the original ideal I .Let { Ω i : 1 ≤ i ≤ s } be the composite divisor sets of the incompatible part ip ( χ ε ) as in Definition 4.7. For a multiplicity i satisfying 1 ≤ i ≤ s and compositedivisor ω i with ω ∈ Ω i ⊂ K [ x ], let us denote ω i as q and consider the normal PQR R q that is isomorphic to the PQR ¯ R = K [ x ] / ( q ) = K [ x ] / ( ω i ) as in Definition5.4. In short, from now on our discussions and computations are over the normalPQR R q as follows. R q ∼ = K [ x ] / ( q ) , q = ω i , ω ∈ Ω i ⊂ K [ x ] . (5.29)We shall follow the pseudo-eliminant computation in Algorithm 3.9 to computethe eliminant of the ideal I + ( q ) = I + ( ω i ) except that we shall compute it overthe normal PQR R q .If we extend the ring epimorphism σ q in (5.4) such that it is the identity mapon the variables ˜ x , then σ q induces a ring epimorphism from ( K [ x ])[ ˜ x ] to R q [ ˜ x ]which we still denote as σ q as follows. σ q : ( K [ x ])[ ˜ x ] → R q [ ˜ x ] : σ q (cid:16) s X j =1 c j ˜ x α j (cid:17) := s X j =1 σ q ( c j ) ˜ x α j . (5.30)Please note that when the composite divisor q bears the form x − a with a ∈ K , the epimorphism σ q in (5.4) becomes σ q ( f ) = f ( a ) ∈ K for f ∈ K [ x ]. Inthis case the coefficients σ q ( c j ) in (5.30) become c j ( a ) ∈ K . We call the inducedepimorphism σ q in (5.30) a specialization associated with a ∈ K .Similarly we can extend the injection ι q in (5.5) to an injection of R q [ ˜ x ] into( K [ x ])[ ˜ x ] in the way that it is the identity map on the variables ˜ x as follows. ι q : R q [ ˜ x ] → ( K [ x ])[ ˜ x ] : ι q (cid:16) s X j =1 c j ˜ x α j (cid:17) := s X j =1 ι q ( c j ) ˜ x α j . (5.31)Further, it is evident that σ q ◦ ι q is the identity map on R q [ ˜ x ]. Lemma 5.19. Let ≻ be an elimination ordering on [ x ] as in Definition 3.1 and F := { f j : 0 ≤ j ≤ s } ⊂ ( K [ x ])[ ˜ x ] \ K a basis of a zero-dimensional ideal I .Suppose that q = ω i is a composite divisor of the incompatible part ip ( χ ε ) as inDefinition 4.7 and R q a normal PQR as in Definition 5.4. Then for F ∩ K [ x ] and G := F \ K [ x ] as in the Initialization of Algorithm 3.9, we have σ q ( F ∩ K [ x ]) = { } and σ q ( G ) is a basis of I q := σ q ( I ) under the epimorphism σ q in (5.30) . roof. The construction of the composite divisor set Ω i in Algorithm 4.6 indicatesthat the pseudo-eliminant χ ε is divisible by the composite divisor q = ω i . Thecomputation of χ ε in Algorithm 3.9 shows that every element of F ∩ K [ x ] isdivisible by f in the Initialization of Algorithm 3.9 and thus by χ ε . Hence F ∩ K [ x ] ⊂ ( ω i ) = ( q ) ⊂ K [ x ], which yields σ q ( F ∩ K [ x ]) = { } . Then readilyfollows the conclusion I q = h σ q ( G ) i with σ q ( G ) ⊂ R q [ ˜ x ] \ R q .In the following Algorithm 5.20 that is parallel to Algorithm 3.9, please notethat all the binary operations over the ring R q in (5.29), i.e., the additions andmultiplications over the ring R q in (5.29), are performed according to those definedin (5.2).Based on Lemma 5.19, in what follows let us abuse the notations a bit andsimply denote σ q ( G ) as F = { f j : 1 ≤ j ≤ s } ⊂ R q [ ˜ x ] \ R q . Algorithm 5.20 (Proper eliminant and proper basis over a normal PQR R q ) . Input: A finite polynomial set F ⊂ R q [ ˜ x ] \ R q .Output: A proper eliminant e q ∈ R q and proper basis B q ⊂ R q [ ˜ x ] \ R q .Initialization: A temporary set S := ∅ in R q [ ˜ x ] \ R q for S -polynomials; atemporary e ∈ R q as e := 0.For each pair f, g ∈ F with f = g , we invoke Procedure R as follows tocompute their S -polynomial S ( f, g ).Procedure R : If lm ( f ) and lm ( g ) are relatively prime, we compute the multiplier σ q ( d ) :=gcd q ( lc ( f ) , lc ( g )) as in (5.20) . If σ q ( d ) ∈ R ∗ q \ R × q , we compute and then addthe S -polynomial S ( f, g ) into the set S . If σ q ( d ) ∈ R × q as in Corollary 5.14,we disregard S ( f, g ) .If there exists an h ∈ F \ { f, g } such that lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i , andthe triangular identity (5.28) has not been applied to the same triplet { f, g, h } before, we compute the multiplier λ as in (5.28) . If λ ∈ R ∗ q \ R × q , we computeand then add the S -polynomial S ( f, g ) into the set S . If λ ∈ R × q , we disregard S ( f, g ) .If neither of the above two cases is true, we compute their S -polynomial S ( f, g ) as in (5.16) . Then we add S ( f, g ) into the set S . End of R We recursively repeat Procedure P as follows for proper reductions of all the S -polynomials in S .Procedure P : For an S ∈ S , we invoke Theorem 5.10 to make a proper reduction of S by F . If the remainder r ∈ R q [ ˜ x ] \ R q , we add r into F . For every f ∈ F \ { r } ,we invoke Procedure R to compute the S -polynomial S ( f, r ) .If the remainder r ∈ R ∗ q \ R × q and e = 0 , we redefine e := σ q (gcd( ι q ( r ) , q )) with σ q and ι q as in (5.4) and (5.5) respectively.If the remainder r ∈ R ∗ q \ R × q and e ∈ R ∗ q , we compute d = gcd q ( r, e ) as in (5.9) . If d is not an associate of e , we redefine e := d .If the remainder r ∈ R × q , we halt the algorithm and output e q = 1 .Then we delete S from S . End of P Q as follows for proper reductions of thespecial kinds of S -polynomials in (5.17) and (5.19).Procedure Q : If S = ∅ and e = 0 , then for every f ∈ F with lc ( f ) ∈ R ∗ q \ R × q , we computethe S -polynomial S ( f, q ) as in (5.19) and add it into S if this has not been donefor f in a previous step.Then we recursively repeat Procedure P .If S = ∅ and e ∈ R ∗ q , then for every f ∈ F with lc ( f ) ∈ R ∗ q \ R × q , if σ q ( d ) := gcd q ( lc ( f ) , e ) ∈ R ∗ q \ R × q as in Corollary 5.14, we compute the S -polynomial S ( f, e ) as in (5.18) and add it into S unless an S -polynomial equalto uS ( f, e ) with u ∈ R × q had been added into S in a previous step.Then we recursively repeat Procedure P . End of Q Finally we define e q := e and B q := F respectively.We output e q and B q . Remark . In Procedure Q of Algorithm 5.20, the condition lc ( f ) ∈ R ∗ q \ R × q in the case of e ∈ R ∗ q is necessary because if lc ( f ) ∈ R × q , we would have d :=gcd q ( lc ( f ) , e ) ∈ R × q as in Corollary 5.14. Definition 5.22 (Proper eliminant e q ; proper basis B q ; modular eliminant χ q ) . We call the standard representation e st q as in (5.6) of the univariate polynomial e q ∈ I q ∩ R q obtained in Algorithm 5.20, whether it is zero or not, a proper eliminant of the ideal I q . In what follows we shall simply denote e q := e st q exceptfor a necessary discrimination in the context. We also call the final polynomialset B q obtained in Algorithm 5.20 a proper basis of I q .Let χ be the eliminant of a zero-dimensional ideal I and q a composite divisoras in (5.29). Suppose that σ q is the epimorphism as in (5.30). Then we define χ q := σ q ( χ ) as the modular eliminant of I q = σ q ( I ). Lemma 5.23. Let χ q and e q be the modular and proper eliminants of the ideal I q respectively as in Definition 5.22. Then the following conclusions hold.(a) If the modular eliminant χ q ∈ R ∗ q , then the eliminant χ is divisible by ι q ( χ st q ) with ι q being the injection as in (5.5) and χ st q the standard representation of χ q in R q as in (5.6) . And we have mult p ( χ ) = mult p ( χ st q ) for every irreduciblefactor p of the composite divisor q = ω i . If χ q = 0 , then χ is divisible by q and we have mult p ( χ ) = mult p ( q ) for every irreducible factor p of q .(b) The epimorphism σ q as in (5.30) is also an epimorphism from I ∩ K [ x ] to I q ∩ R q . Moreover, the proper and modular eliminants e q and χ q of I q satisfy e q ∈ ( χ q ) = I q ∩ R q . In particular, we have e q = 0 if χ q = 0 .(c) For each pair f = g in the polynomial set B q ∪ { e q } with e q ∈ R ∗ q and B q beingthe proper basis of I q , the proper reduction of their S -polynomial S ( f, g ) by B q yields a remainder r ∈ ( e q ) ⊂ ( χ q ) ⊂ R q including the special case of r = 0 .The same holds for the polynomial set B q ∪ { q } when e q = 0 .(d) Algorithm 5.20 terminates in finite steps. roof. Let us first prove (a). When χ q ∈ R ∗ q , by Lemma 5.3 as well as the definitionof the standard representation χ st q of χ q as in (5.6), we know that χ q = uχ st q with u ∈ R × q being a unit. Moreover, for every irreducible factor p of q , we have0 ≤ mult p ( χ st q ) ≤ mult p ( q ) as per Lemma 5.3 (iii) and (5.6). As per Lemma 3.12(i), the pseudo-eliminant χ ε is divisible by χ . By Definition 4.7, the compositedivisor q = ω i satisfies mult p ( q ) = i = mult p ( χ ε ) for every irreducible factor p of q . Hence follows the following inequality:0 < mult p ( χ ) ≤ mult p ( q ) = i (5.32)for every irreducible factor p of q . Based on the division identity χ = hq + ι q ( χ q ) = hq + ι q ( uχ st q ) that is parallel to (5.3), we can deduce that mult p ( χ st q ) =mult p ( ι q ( χ st q )) = mult p ( χ ) for every irreducible factor p of q , which is similar tothe deduction of (5.11). Thus the eliminant χ is divisible by ι q ( χ st q ) as in theconclusion (a) due to the arbitrariness of the irreducible factor p of q .When the modular eliminant χ q = 0, the divisibility of χ by q can be readilydeduced from the definition of the epimorphism σ q in (5.4). Then the equalitymult p ( χ ) = mult p ( q ) for every irreducible factor p of q can be deduced from (5.32).Next let us prove (b). For every r ∈ I q ∩ R q , assume that there exists f ∈ I \ K [ x ] such that σ q ( f ) = r . Then f can be written into f = gq + ι q ( r ) with ι q ( r ) ∈ K [ x ]. Let us denote d := gcd( χ, q ) ∈ K [ x ]. It is evident that we have gqχ/d ∈ h χ i ⊂ I and hence ( f − gq ) χ/d = χ · ι q ( r ) /d ∈ I ∩ K [ x ]. Moreover, σ q ( χ · ι q ( r ) /d ) = rσ q ( χ/d ) such that σ q ( χ/d ) ∈ R × q by Lemma 5.3 (i) since χ/d is relatively prime to q . Thus σ q : I ∩ K [ x ] −→ I q ∩ R q is an epimorphism. Asa result, we have I q ∩ R q = ( χ q ) based on I ∩ K [ x ] = ( χ ) as per Definition 3.3.Then the conclusion (b) readily follows from the fact that e q ∈ I q ∩ R q .The proofs for the conclusions (c) and (d) are almost verbatim repetitions ofthose for Lemma 3.12 (ii) and (iii). In particular, the argument for (d) is basedon the fact that R q [ ˜ x ] is also a Noetherian ring. In fact, the normal PQR R q inDefinition 5.4 is isomorphic to the Noetherian PQR ¯ R in Definition 5.1. Corollary 5.24. If the proper eliminant e q ∈ R ∗ q , then the eliminant χ is notdivisible by the composite divisor q = ω i of the incompatible part ip ( χ ε ) . Moreover,if the proper eliminant e q ∈ R × q , then the eliminant χ is relatively prime to thecomposite divisor q = ω i .Proof. If the eliminant χ is divisible by the composite divisor q , then the modulareliminant χ q = σ q ( χ ) = 0. By Lemma 5.23 (b) we can deduce that e q = 0,contradicting e q ∈ R ∗ q .If the proper eliminant e q ∈ R × q , there exists b ∈ R × q such that 1 = b e q ∈ ( χ q )since we have e q ∈ ( χ q ) by Lemma 5.23 (b). Hence the modular eliminant χ q = σ q ( χ ) ∈ R × q , from which we can deduce that the eliminant χ is relatively prime tothe composite divisor q = ω i by Lemma 5.3 (i).In what follows let us exclude the trivial case when the proper eliminant e q ∈ R × q . That is, let us assume that the eliminant χ is not relatively prime to thecomposite divisor q = ω i . 33 emma 5.25. Let F = { f j : 1 ≤ j ≤ s } ⊂ R q [ ˜ x ] \ R q be a polynomial set over anormal PQR R q as in (5.29) . Suppose that for ≤ j ≤ s , each f j has the sameleading monomial lm ( f j ) = ˜ x α ∈ [ ˜ x ] .(a) If f = P sj =1 f j satisfies lm ( f ) ≺ ˜ x α , then there exist multipliers b, b j ∈ R ∗ q for ≤ j < s such that bf = X ≤ j Suppose that χ is the eliminant of a zero-dimensional ideal I in ( K [ x ])[ ˜ x ] over a perfect field K and χ ε a pseudo-eliminant of I . Let q = ω i bea composite divisor of the incompatible part ip ( χ ε ) and R q the normal PQR as in (5.29) . Let e q and χ q in R q denote the proper and modular eliminants respectivelyas in Definition 5.22.(a) If the proper eliminant e q = 0 , then the eliminant χ is divisible by thecomposite divisor q = ω i of the incompatible part ip ( χ ε ) and hence the modulareliminant χ q = 0 . For every irreducible factor p of the composite divisor q , wehave mult p ( χ ) = i .(b) If the proper eliminant e q ∈ R ∗ q , then the eliminant χ is divisible by ι q ( e q ) with ι q being the injection as in (5.5) and e q = e st q as in Definition 5.22. Hencethe modular eliminant χ st q = e q . For every irreducible factor p of the compositedivisor q , we have mult p ( χ ) = mult p ( e q ) .Proof. Let us fix an irreducible factor p of the composite divisor q = ω i . If F is the originally given basis of the ideal I in ( K [ x ])[ ˜ x ], then σ q ( F ) is a basisof the ideal I q = σ q ( I ) in R q under the epimorphism σ q in (5.30) according toLemma 5.19. For simplicity let us abuse the notation a bit and still denote σ q ( F )as F = { f j : 1 ≤ j ≤ s } ⊂ R q [ ˜ x ] \ R q . Then there exist h j ∈ R q [ ˜ x ] for 1 ≤ j ≤ s such that the modular eliminant χ q = σ q ( χ ) ∈ R q can be written as: χ q = s X j =1 h j f j . (5.38)Suppose that max ≤ j ≤ s { lm ( h j ) · lm ( f j ) } = ˜ x β . Similar to (4.9), we collect andrename the set { f j : lm ( h j f j ) = ˜ x β , ≤ j ≤ s } as a new set B t := { g j : 1 ≤ j ≤ t } .Let us first make an assumption that B t = ∅ . We shall address the special case35f B t = ∅ shortly afterwards. Of course the subscripts of the functions { h j } arerelabelled accordingly. It is easy to see that for g j ∈ B t with 1 ≤ j ≤ t , we have lc ( h j ) · lc ( g j ) ∈ R ∗ q . In this way (5.38) can be written as: χ q = t X j =1 h j g j + X f j ∈ F \ B t h j f j , (5.39)where the products h j f j are those in (5.38) with f j ∈ F \ B t for 1 ≤ j ≤ s .With lt ( h j ) := c j ˜ x α j and lc ( h j ) = c j ∈ R ∗ q for 1 ≤ j ≤ t , it suffices to studythe following polynomial that is a summand of (5.39): g := t X j =1 lt ( h j ) · g j = t X j =1 c j ˜ x α j g j . (5.40)From lm ( χ q ) = 1 ≺ ˜ x β in (5.39) we can deduce that lm ( g ) ≺ ˜ x β in (5.40). Asper Lemma 5.25 (a), there exist multipliers b, b j ∈ R ∗ q for 1 ≤ j < t such that: bg = X ≤ j In this section we procure the exact form of the eliminant χ of a zero-dimensionalideal I . This is based on our former analyses of the pseudo-eliminant χ ε of I , i.e.,of its compatible part cp ( χ ε ) in Theorem 4.10 and incompatible part ip ( χ ε ) inTheorem 5.26 respectively. We also formulate a decomposition of I according to cp ( χ ε ) and the composite divisors of ip ( χ ε ). In this way we acquire a new type ofbases for I based on the exact form of χ , pseudo-basis B ε obtained in Algorithm3.9 and proper bases B q obtained in Algorithm 5.20. Moreover, we address theideal membership problem for this new type of bases and characterize the newtype of bases in terms of their leading terms. Definition 6.1 (Proper divisors θ q and proper factor χ ip ) . For every composite divisor q of the incompatible part ip ( χ ε ) as in Definition4.7, there corresponds to a proper eliminant e q as in Definition 5.22. We define a proper divisor θ q ∈ K [ x ] in accordance with e q as follows.If e q ∈ R × q , we define θ q := 1;If e q = 0, we define θ q := q ;If e q ∈ R ∗ q \ R × q , we define θ q := ι q ( e q ) with ι q being the injection as in (5.5)and e q = e st q as in Definition 5.22.We say that a proper divisor θ q ∈ K [ x ] is nontrivial if θ q = 1.We define the proper factor χ ip as the product of all the proper divisors θ q . Theorem 6.2. The eliminant χ of a zero-dimensional ideal I is the product ofthe compatible part cp ( χ ε ) of the pseudo-eliminant χ ε and proper factor χ ip as inDefinition 6.1. That is, χ = cp ( χ ε ) · χ ip . Moreover, the compatible part cp ( χ ε ) isrelatively prime to the proper factor χ ip .Proof. According to Lemma 3.12 (i), every irreducible factor p of the eliminant χ is also a factor of either the compatible part cp ( χ ε ) or a composite divisor q = ω i of the incompatible part ip ( χ ε ) as per (4.1). Moreover, the proper factor χ ip isdefined as the product of the proper divisors θ q for all the composite divisors q of the incompatible part ip ( χ ε ). Hence we can easily deduce the conclusion fromTheorem 4.10 and Theorem 5.26 as well as the definition of proper divisors θ q inDefinition 6.1. Finally, cp ( χ ε ) and χ ip are relatively prime since cp ( χ ε ) and ip ( χ ε )are relatively prime.The nontrivial proper divisors θ q in Definition 6.1 as well as the compatiblepart cp ( χ ε ) of the pseudo-eliminant χ ε are pairwise relatively prime. In fact, thecomposite divisor q is divisible by the proper divisor θ q = ι q ( e st q ) when the propereliminant e q ∈ R ∗ q \ R × q in Definition 6.1. And θ q = q when e q = 0. The compositedivisors q are pairwise relatively prime as in (4.1). In this way the nontrivialproper divisors θ q and the compatible part cp ( χ ε ) constitute a factorization ofthe eliminant χ according to Theorem 6.2. In the following lemma we make adecomposition of the zero-dimensional ideal I in accordance with the factorizationof the eliminant χ . Please also refer to [BW93, P337, Lemma 8.5] for a similar decomposition. emma 6.3. Let { b j : 1 ≤ j ≤ s } ⊂ K [ x ] \ K be pairwise relatively prime and b := Q sj =1 b j . Then for an arbitrary ideal J ⊂ ( K [ x ])[ ˜ x ] , we have: J + h b i = s \ j =1 ( J + h b j i ) , (6.1) where h b i and h b j i denote the principal ideals in ( K [ x ])[ ˜ x ] generated by b and b j respectively for ≤ j ≤ s .Proof. It is evident that the inclusion “ ⊂ ” holds. The proof of the reverse inclusionis as follows. For every f ∈ T sj =1 ( J + h b j i ), there exist g j ∈ J and h j ∈ ( K [ x ])[ ˜ x ]such that f = g j + h j b j for 1 ≤ j ≤ s . Moreover, { b/b j : 1 ≤ j ≤ s } have nonontrivial common factors. Hence there exist a j ∈ K [ x ] for 1 ≤ j ≤ s such that P sj =1 a j b/b j = 1. Now we have: f = s X j =1 f a j b/b j = s X j =1 ( g j + h j b j ) a j b/b j = s X j =1 a j bb j g j + b s X j =1 h j a j ∈ J + h b i . Let Θ := { q j : 1 ≤ j ≤ t } be the set of composite divisors of the incompatiblepart ip ( χ ε ) such that their corresponding proper divisors θ q j as in Definition 6.1are not trivial, i.e., θ q j = 1 for 1 ≤ j ≤ t . We have the following decomposition ofthe zero-dimensional ideal I = I + h χ i according to Theorem 6.2 and Lemma 6.3. I = ( I + h cp ( χ ε ) i ) ∩ \ q ∈ Θ ( I + h θ q i ) . (6.2) Lemma 6.4. Suppose that I ⊂ ( K [ x ])[ ˜ x ] is a zero-dimensional ideal with anelimination ordering on [ x ] as in Definition 3.1. Let B ε = { g k : 1 ≤ k ≤ s } be apseudo-basis of I and d = cp ( χ ε ) the compatible part of the pseudo-eliminant χ ε associated with B ε . For every f ∈ I , there exist { v k : 0 ≤ k ≤ τ } ⊂ ( K [ x ])[ ˜ x ] and a multiplier λ relatively prime to d such that: λf = τ X k =1 v k g k + v χ ε . (6.3) Moreover, the polynomials in (6.3) satisfy the following condition: lm ( f ) = max (cid:8) max ≤ k ≤ τ { lm ( v k g k ) } , lm ( v ) (cid:9) . (6.4) Proof. The proof for the conclusion is almost a verbatim repetition of that forTheorem 4.10. More specifically, suppose that f ∈ I can be written as f = s X j =0 h j f j (6.5)with { f j : 0 ≤ j ≤ s } ⊂ ( K [ x ])[ ˜ x ] \ K being the basis G ∪ { f } of the ideal I after the Initialization in Algorithm 3.9 and { h j : 0 ≤ j ≤ s } ⊂ ( K [ x ])[ ˜ x ]. Inparticular, f ∈ ( χ ε ) ⊂ ( d ) ⊂ K [ x ] \ K ∗ . It is evident that the conclusion holdswhen lm ( f ) = max ≤ j ≤ s { lm ( h j f j ) } . 42o in what follows let us suppose that lm ( f ) ≺ max ≤ j ≤ s { lm ( h j f j ) } . In thiscase we treat f as the eliminant χ in (4.8). Let us fix an irreducible factor p ofthe compatible part d = cp ( χ ε ). We repeat the arguments verbatim from (4.9)through (4.23) to obtain a new representation like (4.23) as follows. bλf = τ X k =1 µ k g k + µ χ ε (6.6)such that the multiplier bλ is relatively prime to p , i.e., mult p ( bλ ) = 0. Here { g k : 1 ≤ k ≤ τ } = B ε is the pseudo-basis obtained in Algorithm 3.9 and µ k ∈ ( K [ x ])[ ˜ x ] for 0 ≤ k ≤ τ . The leading monomials of the representation in (6.6)are strictly less than those in (6.5), which is similar to (4.24). We repeat thisprocedure of rewriting the representations of bλf so that their leading monomialsstrictly decrease. Moreover, the multipliers for the representations are alwaysrelatively prime to the irreducible factor p of the compatible part d . Since theelimination ordering on ( K [ x ])[ ˜ x ] is a well-ordering and the leading monomialsof a representation of f cannot be strictly less than lm ( f ), after a finite numberof repetitions we obtain a representation bearing the following form: νf = τ X k =1 w k g k + w χ ε (6.7)with w k ∈ ( K [ x ])[ ˜ x ] for 0 ≤ k ≤ τ such thatmax (cid:8) max ≤ k ≤ τ { lm ( w k g k ) } , lm ( w χ ε ) (cid:9) = lm ( f ) . (6.8)Moreover, the multiplier ν ∈ ( K [ x ]) ∗ in (6.7) is relatively prime to the irreduciblefactor p of the compatible part d = cp ( χ ε ).Suppose that the compatible part d has a factorization into a product ofcompatible divisors that are pairwise relatively prime as in Definition 4.5, i.e., d = Q tl =1 p n l l with n l ∈ N ∗ . For each irreducible factor p l of d , there correspondsto a representation of f in (6.7) that can be indexed by the subscript l of p l with1 ≤ l ≤ t as follows. ν l f = τ X k =1 w ( l ) k g k + w ( l )0 χ ε , (6.9)where the multiplier ν l ∈ ( K [ x ]) ∗ in (6.9) is relatively prime to the irreduciblefactor p l of the compatible part d . Moreover, the leading monomial identity (6.8)still holds for w ( l ) k and w ( l )0 with 1 ≤ l ≤ t in (6.9), i.e.,max (cid:8) max ≤ k ≤ τ { lm ( w ( l ) k g k ) } , lm ( w ( l )0 χ ε ) (cid:9) = lm ( f ) . (6.10)Let us denote λ := gcd ≤ l ≤ t { ν l } ∈ ( K [ x ]) ∗ . Then λ is relatively prime to thecompatible part d . There exist { u l ∈ K [ x ] : 1 ≤ l ≤ t } such that λ = P tl =1 u l ν l .Hence we can obtain a representation of f as follows. λf = t X l =1 u l ν l f = τ X k =1 g k t X l =1 u l w ( l ) k + χ ε t X l =1 u l w ( l )0 := τ X k =1 v k g k + v χ ε (6.11)43ith v k := P tl =1 u l w ( l ) k for 0 ≤ k ≤ τ in ( K [ x ])[ ˜ x ]. This is (6.3) proved.Based on the identities v k = P tl =1 u l w ( l ) k in (6.11) for 0 ≤ k ≤ τ , we can inferthe following inequalities between their leading monomials: lm ( v χ ε ) (cid:22) max ≤ l ≤ t { lm ( w ( l )0 χ ε ) } ; lm ( v k g k ) (cid:22) max ≤ l ≤ t { lm ( w ( l ) k g k ) } (6.12)for 1 ≤ k ≤ τ since u l ∈ K [ x ] for 1 ≤ l ≤ t . A combination of (6.12) and (6.10)leads to: max (cid:8) max ≤ k ≤ τ { lm ( v k g k ) } , lm ( v χ ε ) (cid:9) (cid:22) lm ( f ) . (6.13)We can also infer the reverse inequality of (6.13) from (6.11). Thus follows theequality (6.4).The following intriguing observation is crucial for its ensuing conclusions. Lemma 6.5. Suppose that I ⊂ ( K [ x ])[ ˜ x ] is a zero-dimensional ideal with anelimination ordering on [ x ] as in Definition 3.1. Let d = cp ( χ ε ) be the compatiblepart of a pseudo-eliminant χ ε of I . Consider the epimorphism σ d : ( K [ x ])[ ˜ x ] → R d [ ˜ x ] as in (5.30) such that I d := σ d ( I ) . If we define I ∗ := { f ∈ I : σ d ( lc ( f )) ∈ R ∗ d } , then σ d ( I ∗ ) = I d .Proof. Let χ ip be the proper factor as in Definition 6.1 such that d · χ ip = χ with χ being the eliminant of I as per Theorem 6.2. For every h ∈ I d with lc ( h ) ∈ R ∗ d ,suppose that σ d ( g ) = h with g ∈ I . Then g − ι d ( h ) ∈ h d i with the injection ι d defined like in (5.31). Hence χ ip g − χ ip · ι d ( h ) ∈ h χ i ⊂ I . Thus χ ip · ι d ( h ) ∈ I and σ d ( χ ip · ι d ( h )) = σ d ( χ ip ) · σ d ( ι d ( h )) = σ d ( χ ip ) · h since σ d ◦ ι d is the identity map. Wecan further infer that σ d ( χ ip ) is a unit in R d by Lemma 5.3 (i) since χ ip is relativelyprime to d as per Theorem 6.2. Hence σ d ( χ ip ) · lc ( h ) ∈ R ∗ d , from which we candeduce that χ ip · ι d ( h ) ∈ I ∗ . From σ d ( χ ip ) ∈ R × d and σ d ( χ ip ) · h = σ d ( χ ip · ι d ( h )),we can also deduce that I d = σ d ( χ ip ) · I d := { σ d ( χ ip ) · h : h ∈ I d } ⊂ σ d ( I ∗ ). Theinclusion I d = σ d ( I ) ⊃ σ d ( I ∗ ) is evident.For the ideal I + h d i = I + h cp ( χ ε ) i in (6.2), we provide a characterization ofthe basis B ε ∪ { d } in the following conclusions with B ε being a pseudo-basis of I as in Definition 3.11. In particular, we address the ideal membership problem forthe ideal I + h d i . Lemma 6.6. Suppose that I ⊂ ( K [ x ])[ ˜ x ] is a zero-dimensional ideal with anelimination ordering on [ x ] as in Definition 3.1. Let B ε = { g k : 1 ≤ k ≤ s } be apseudo-basis of I and d = cp ( χ ε ) the compatible part of the pseudo-eliminant χ ε associated with B ε . Consider the epimorphism σ d as follows. σ d : ( K [ x ])[ ˜ x ] −→ R d [ ˜ x ] (6.14) is defined like in (5.30) such that I d := σ d ( I ) and B d := σ d ( B ε ) . Then with anelimination ordering on R d [ ˜ x ] like in Definition 5.7, we have the following idealidentity in R d [ ˜ x ] : h lt ( I d ) i = h lt ( B d ) i . (6.15)44 roof. For every g ∈ I d , let f ∈ I ∗ as in Lemma 6.5 such that σ d ( f ) = g and inparticular, σ d ( lc ( f )) = lc ( g ) ∈ R ∗ d . According to Lemma 6.4, there exist { v k : 0 ≤ k ≤ s } ⊂ ( K [ x ])[ ˜ x ] as well as a multiplier λ ∈ K [ x ] that is relatively prime to d = cp ( χ ε ) such that both (6.3) and (6.4) hold for this f ∈ I ∗ . We apply theepimorphism σ d like in (5.30) to the identity (6.3). Then σ d ( λ ) ∈ R × d by Lemma5.3 (i) since λ is relatively prime to d . We have σ d ( lt ( λf )) = σ d ( λ ) · lt ( g ) = 0since σ d ( lc ( f )) = lc ( g ) ∈ R ∗ d whereas σ d ( lt ( v χ ε )) = σ d ( χ ε · lt ( v )) = 0 since χ ε ∈ ( d ) ⊂ K [ x ]. For 1 ≤ k ≤ s , we collect the subscript k into a set Λ if lm ( v k ) · lm ( g k ) = lm ( g ) = lm ( f ) and σ d ( lc ( v k g k )) = σ d ( lc ( v k ) · lc ( g k )) ∈ R ∗ d .Then based on (6.4) we have Λ = ∅ since σ d ( lt ( λf )) = 0 on the left hand sideof (6.3). In the case of σ d ( lc ( g k )) ∈ R ∗ d for k ∈ Λ, we also have σ d ( lt ( g k )) = lt ( σ d ( g k )). Hence the following identity: lt ( g ) = σ d ( λ ) − · X k ∈ Λ σ d ( lt ( v k )) · lt ( σ d ( g k )) ∈ h lt ( B d ) i (6.16)indicates the ideal identity (6.15). Lemma 6.7. Let R q be a normal PQR as in Definition 5.4. Suppose that { c j : 0 ≤ j ≤ s } ⊂ R ∗ q . There exist { b j : 1 ≤ j ≤ s } ⊂ R q such that c = P sj =1 b j c j if andonly if c ∈ ( c ) ⊂ R q with c := gcd( { c j : 1 ≤ j ≤ s } ) as in (5.8) or (5.9) . Inparticular, for every proper subset Λ ⊂ { ≤ j ≤ s } , the identity c = P j ∈ Λ d j c j holds with { d j : j ∈ Λ } ⊂ R q only if c ∈ ( c ) .Proof. First of all, there exist { a j : 1 ≤ j ≤ s } ⊂ R q such that c = s X j =1 a j c j . (6.17)In fact, let ι q be the injection defined in (5.5). If d := gcd( { ι q ( c j ) : 1 ≤ j ≤ s } ),then there exist { d j : 1 ≤ j ≤ s } ⊂ K [ x ] such that d = P sj =1 d j · ι q ( c j ). Applying σ q to this identity, we have σ q ( d ) = s X j =1 σ q ( d j ) · c j (6.18)since σ q ◦ ι q is the identity map. Hence σ q ( d ) ∈ ( c ) since c j ∈ ( c ) for 1 ≤ j ≤ s .On the other hand, ι q ( c j ) ∈ ( d ) for 1 ≤ j ≤ s . Thus c j ∈ ( σ q ( d )) for 1 ≤ j ≤ s and hence their common divisor c ∈ ( σ q ( d )). By the standard representations of c and σ q ( d ) as in (5.6), we have c = u · σ q ( d ) with u ∈ R × q . As a result, it sufficesto take a j := u · σ q ( d j ) for 1 ≤ j ≤ s in (6.18) in order to deduce (6.17).Now the necessity of the conclusion is easy to verify. And the sufficiency ofthe conclusion readily follows from (6.17). Notation . Suppose that B ⊂ R q [ ˜ x ] \ R q is a finite polynomial set and ˜ x α ∈ [ ˜ x ] .We denote B | ˜ x α := { b ∈ B : ˜ x α ∈ h lm ( b ) i} .Hereafter we shall simply use gcd to denote the gcd st in (5.8) or gcd q in (5.9).The two definitions only differ by a unit as in (5.10), which has no impact on ourconclusions. 45t is evident that B | ˜ x α = ∅ is equivalent to ˜ x α ∈ h lm ( B ) i since Lemma 2.1applies to PQR as well. Please note that this is also the condition for a nonzeroterm c α ˜ x α to be pseudo-reducible with respect to B in Definition 2.6. Definition 6.9 ( gcd -reducible terms and polynomials in R q [ ˜ x ]) . Let ρ ∈ R ∗ q and B ⊂ R q [ ˜ x ] \ R q be a finite polynomial set. We say that a nonzeroterm c α ˜ x α is gcd -reducible with respect to B if B | ˜ x α = ∅ and c α ∈ ( d ) ⊂ R q with d := gcd( { lc ( b ) : b ∈ B | ˜ x α } ) as in Notation 6.8. We also say that c α ˜ x α is gcd -reduced with respect to B if it is not gcd -reducible with respect to B .A polynomial f ∈ R q [ ˜ x ] \ R q is said to be gcd -reducible with respect to B if f has a gcd -reducible term. Otherwise f is said to be gcd -reduced with respectto B . Lemma 6.10. Let B ⊂ R q [ ˜ x ] \ R q be a finite polynomial set. Then a nonzeroterm c α ˜ x α ∈ h lt ( B ) i if and only if c α ˜ x α is gcd -reducible with respect to B .Proof. We only prove the necessity condition since the sufficiency condition isevident by Lemma 6.7. Suppose that lt ( B ) = { a j ˜ x α j : 1 ≤ j ≤ s } and c α ˜ x α = P sj =1 a j h j ˜ x α j with h j ∈ R q [ ˜ x ] for 1 ≤ j ≤ s . We expand each h j into individualterms and compare the terms with the same monomial ˜ x α on both sides of theequality. In this way we obtain an equality c α ˜ x α = P sj =1 c β j a j ˜ x α j ˜ x β j instead with c β j ˜ x β j being a term of h j such that α j + β j = α for 1 ≤ j ≤ s . Now it is evidentthat B | ˜ x α = ∅ and c α = P sj =1 a j c β j . Then the conclusion readily follows fromLemma 6.7. Definition 6.11 ( gcd -term reduction in R q [ ˜ x ]) . Suppose that f ∈ R q [ ˜ x ] \ R q has a term c α ˜ x α that is gcd -reducible withrespect to a finite set B ⊂ R q [ ˜ x ] \ R q . Let us denote d := gcd( { lc ( b ) : b ∈ B | ˜ x α } )as in Notation 6.8. With l α := ι q ( c α ) and l d := ι q ( d ), let us define the multipliers µ := σ q (lcm( l α , l d ) /l α ) and m := σ q (lcm( l α , l d ) /l d ). Then we can make a gcd -termreduction of f by B as follows. h = µf − X b ∈ B | ˜ x α mc b x α lm ( b ) b, (6.19)where d = P b ∈ B | ˜ x α c b · lc ( b ) with c b ∈ R q . We call h the remainder of the reductionand µ the interim multiplier on f with respect to B . Theorem 6.12 ( gcd -division in R q [ ˜ x ]) . With the elimination ordering on R q [ ˜ x ] as in Definition 5.7, suppose that B = { g j : 1 ≤ j ≤ s } ⊂ R q [ ˜ x ] \ R q is a polynomial set. For every f ∈ R q [ ˜ x ] , there exista multiplier λ ∈ R × q as well as a remainder r ∈ R q [ ˜ x ] and quotients h j ∈ R q [ ˜ x ] for ≤ j ≤ s such that λf = s X j =1 h j g j + r, (6.20) where r is gcd -reduced with respect to B . Moreover, the polynomials in (6.20) satisfy the following condition: lm ( f ) = max (cid:8) max ≤ j ≤ s { lm ( h j ) · lm ( g j ) } , lm ( r ) (cid:9) . (6.21)46 roof. Similar to the proof of Theorem 5.10 for proper division, the proof is almosta verbatim repetition of that for Theorem 2.7 if we substitute R q [ ˜ x ] for ( K [ x ])[ ˜ x ].In fact, the maximal term of h that is gcd -reducible with respect to B is strictlyless than that of f after we make a gcd -term reduction as in (6.19). Moreover,it suffices to prove that the condition (6.21) applies to the gcd -term reduction in(6.19), same as in the proof of Theorem 2.7.We also call the gcd -division with respect to B a gcd -reduction with respectto B henceforth. Theorem 6.13. Suppose that I ⊂ ( K [ x ])[ ˜ x ] is a zero-dimensional ideal with anelimination ordering on [ x ] as in Definition 3.1. Let B ε = { g k : 1 ≤ k ≤ s } be apseudo-basis of I and d = cp ( χ ε ) the compatible part of the pseudo-eliminant χ ε associated with B ε . Let σ d : ( K [ x ])[ ˜ x ] → R d [ ˜ x ] be the epimorphism as in (5.30) such that I d := σ d ( I ) and B d := σ d ( B ε ) . Then the identity (6.15) in Lemma 6.6is equivalent to a characterization of B d as following:For every f ∈ R d [ ˜ x ] , we have f ∈ I d if and only if we can make a gcd -reduction of f with respect to B d as in (6.20) and (6.21) such that the remainder r = 0 .Proof. Suppose that the identity (6.15) holds. For every f ∈ I d , we make a gcd -reduction of f with respect to B d as in (6.20) and (6.21) such that the remainder r is gcd -reduced with respect to B d as in Definition 6.9. On the other hand, theremainder r ∈ I d as per the expression in (6.20). If r = 0, there exist { h k : 1 ≤ k ≤ s } ⊂ R d [ ˜ x ] such that lt ( r ) = P sk =1 h k · lt ( σ d ( g k )) according to (6.15). For 1 ≤ k ≤ s , we collect the subscript k into a set Λ if h k has a term denoted as c k ˜ x α k thatsatisfies ˜ x α k · lm ( σ d ( g k )) = lm ( r ) and c k · lc ( σ d ( g k )) ∈ R ∗ d . Then we have Λ = ∅ since lc ( r ) ∈ R ∗ d . Moreover, lc ( r ) = P k ∈ Λ c k · lc ( σ d ( g k )) and hence lc ( r ) ∈ ( a )by Lemma 6.7 with a := gcd( { lc ( σ d ( g k )) : k ∈ Λ } ). According to Definition 6.9, r is gcd -reducible with respect to B d , which constitutes a contradiction. Thisproves the necessity of the conclusion. The sufficiency of the conclusion is evidentsince B d ⊂ I d .Next let us prove the identity (6.15) under the assumption that every f ∈ I d canbe gcd -reduced to r = 0 by B d . In fact, f = P sk =1 q k · σ d ( g k ) with q k ∈ R d [ ˜ x ] suchthat lm ( f ) = max ≤ k ≤ s { lm ( q k ) · lm ( σ d ( g k )) } according to (6.21). For 1 ≤ k ≤ s ,we collect the subscript k into a set Λ if lm ( q k ) · lm ( σ d ( g k )) = lm ( f ). Then lt ( f ) = P k ∈ Λ lt ( q k ) · lt ( σ d ( g k )). Thus lt ( I d ) ⊂ h lt ( B d ) i . The other directionof (6.15) is trivial to prove.In what follows we provide a modular characterization of the ideal I + h θ q i in (6.2) over the normal PQR R q . We shall use a modular argument for thecharacterization resorting to the proper basis B q and proper eliminant e q obtainedin Algorithm 5.20. Lemma 6.14. Suppose that χ ε is a pseudo-eliminant of a zero-dimensional ideal I with q being a composite divisor of its incompatible part ip ( χ ε ) . Let e q and B q = { g k : 1 ≤ k ≤ τ } be the proper eliminant and proper basis of I q = σ q ( I ) respectively with σ q being the epimorphism as in (5.30) . For every f ∈ I q , there xist a multiplier λ ∈ R × q and { v k : 0 ≤ k ≤ τ } ⊂ R q [ ˜ x ] such that: λf = τ X k =1 v k g k + v e q . (6.22) Moreover, the polynomials in (6.22) satisfy the following condition: lm ( f ) = max (cid:8) max ≤ k ≤ τ { lm ( v k ) · lm ( g k ) } , lm ( v ) (cid:9) . (6.23) In particular, the above conclusions are still sound when the proper eliminant e q = 0 .Proof. The proof is an almost verbatim repetition of that for Theorem 5.26, whichis similar to the proof for Lemma 6.4. In fact, suppose that F is the originally givenbasis of the ideal I in ( K [ x ])[ ˜ x ] such that σ q ( F ) = { f j : 1 ≤ j ≤ s } ⊂ R q [ ˜ x ] \ R q is a basis of the ideal I q = σ q ( I ). Then for every f ∈ I q , there exist h j ∈ R q [ ˜ x ] for1 ≤ j ≤ s such that f can be written as: f = s X j =1 h j f j . (6.24)Thus the conclusion readily follows when lm ( f ) = max ≤ j ≤ s { lm ( h j ) · lm ( f j ) } since σ q ( F ) ⊂ B q .Now let us suppose that lm ( f ) ≺ max ≤ j ≤ s { lm ( h j ) · lm ( f j ) } . In this case wetreat f as the modular eliminant χ q in (5.38). Let us fix an irreducible factor p ofthe composite divisor q . We repeat the arguments verbatim from (5.39) through(5.53) to obtain a new representation like in (5.53) as follows. bλf = τ X k =1 µ k g k + µ e q (6.25)with µ k ∈ R q [ ˜ x ] for 0 ≤ k ≤ τ . The leading monomials of the representationin (6.25) are strictly less than those in (6.24), which resembles (5.54) closely.Moreover, the multiplier bλ in (6.25) is relatively prime to the irreducible factor p ofthe composite divisor q , i.e., mult p ( bλ ) = 0. We repeat this procedure of rewritingthe representations of bλf so that their leading monomials strictly decrease. Andthe multipliers for the representations are always relatively prime to the irreduciblefactor p of the composite divisor q . After a finite number of repetitions we shallobtain a representation in the following form: νf = τ X k =1 w k g k + w e q (6.26)with w k ∈ R q [ ˜ x ] for 0 ≤ k ≤ τ such thatmax (cid:8) max ≤ k ≤ τ { lm ( w k ) · lm ( g k ) } , lm ( w ) (cid:9) = lm ( f ) . (6.27)The multiplier ν ∈ R ∗ q in (6.26) is relatively prime to the irreducible factor p ofthe composite divisor q . 48ince for every irreducible factor p of the composite divisor q , we have (6.26)and (6.27), we can repeat the arguments almost verbatim in (6.9) and (6.11) toshow that there exist a multiplier λ ∈ R × q and { v k : 0 ≤ k ≤ τ } ⊂ R q [ ˜ x ] suchthat (6.22) holds. In the arguments we substitute the proper eliminant e q for thepseudo-eliminant χ ε and use (6.17) for the representation of a greatest commondivisor in R q . Moreover, we can corroborate (6.23) by repeating almost verbatimthe arguments in (6.10), (6.12) and (6.13). In fact, it suffices to substitute lm ( w ( l ) k ) · lm ( g k ) for lm ( w ( l ) k g k ) in (6.10) and (6.12), as well as to substitute lm ( v k ) · lm ( g k )for lm ( v k g k ) in (6.12) and (6.13), as regards the existence of zero divisors in R q .Let I ⊂ ( K [ x ])[ ˜ x ] be a zero-dimensional ideal. For a composite divisor q of theincompatible part ip ( χ ε ) of a pseudo-eliminant χ ε of I , let R q be the normal PQRdefined as in (5.29). Suppose that e q ∈ R ∗ q \ R × q is the proper eliminant obtainedin Algorithm 5.20. In particular, let e q stand for the standard representation e st q as in Definition 5.22. As per (5.6), we have q ∈ ( ι q ( e q )) ⊂ K [ x ] with ι q being theinjection defined as in (5.5). Hence similar to (5.1), let us define: R p := { r ∈ R q : deg( r ) < deg( e q ) } . (6.28)Similar to (5.2), we can redefine the binary operations on R p such that R p is anormal PQR satisfying R p ∼ = R q / ( e q ). For every f ∈ R q , there exist a quotient h ∈ R q and unique remainder r ∈ R q satisfying f = h e q + r such that deg( r ) < deg( e q ). Like in (5.4) we can define an epimorphism σ p : R q → R p as σ p ( f ) := r .This combined with (5.3) and (5.4) lead to an epimorphism σ p ◦ σ q : K [ x ] → R p .For every f ∈ K [ x ], there exist a quotient h ∈ K [ x ] and unique remainder r ∈ K [ x ] such that f = h · ι q ( e q ) + r . Hence we can also define an epimorphism˜ σ p : K [ x ] → R p as ˜ σ p ( f ) := r . Since q ∈ ( ι q ( e q )), it is evident that ˜ σ p = σ p ◦ σ q .For simplicity we still denote ˜ σ p as σ p henceforth.Similar to (5.30), σ p can be extended to a ring epimorphism from ( K [ x ])[ ˜ x ]or R q [ ˜ x ] to R p [ ˜ x ] which we still denote as σ p as follows. σ p : ( K [ x ])[ ˜ x ] or R q [ ˜ x ] → R p [ ˜ x ] : σ p (cid:16) s X j =1 c j ˜ x α j (cid:17) := s X j =1 σ p ( c j ) ˜ x α j . (6.29)Similar to (5.31), we also have an injection ι p as follows. ι p : R p [ ˜ x ] → ( K [ x ])[ ˜ x ] or R q [ ˜ x ] : ι p (cid:16) s X j =1 c j ˜ x α j (cid:17) := s X j =1 ι p ( c j ) ˜ x α j . (6.30) Theorem 6.15. Suppose that I ⊂ ( K [ x ])[ ˜ x ] is a zero-dimensional ideal over aperfect field K and χ ε a pseudo-eliminant of I . Let q be a composite divisor of theincompatible part ip ( χ ε ) and R q the normal PQR as in (5.29) . Let e q ∈ R q \ R × q and B q denote the proper eliminant and proper basis obtained in Algorithm 5.20respectively.If e q = 0 , then we have the following two equivalent characterizations of B q :(1) A characterization of B q through its leading terms via an ideal identity asfollows. h lt ( I q ) i = h lt ( B q ) i . (6.31)49 2) A characterization of B q through gcd -reductions:For every f ∈ R q [ ˜ x ] , we have f ∈ I q if and only if we can make a gcd -reduction of f with respect to B q as in (6.20) and (6.21) with the remainder r = 0 .If e q ∈ R ∗ q \ R × q , we define I p := σ p ( I q ) = σ p ( I ) and B p := σ p ( B q ) = σ p ( B ε ) with σ p being the epimorphism as in (6.29) . Then we have the following two equivalentcharacterizations of B p :(3) A characterization of B p through its leading terms via an ideal identity asfollows. h lt ( I p ) i = h lt ( B p ) i . (6.32) (4) A characterization of B p through gcd -reductions:For every f ∈ R p [ ˜ x ] , we have f ∈ I p if and only if we can make a gcd -reduction of f with respect to B p as in (6.20) and (6.21) with the remainder r = 0 .Proof. The identities (6.31) and (6.32) follow directly from Lemma 6.14. Theproofs are similar to and even simpler than that for the identity (6.15) in Lemma6.6 since we need a conclusion like Lemma 6.5 in neither cases. In fact, we canobtain (6.31) as an identity of leading terms from the identity (6.22). We firstdefine a subscript set Λ for (6.22) as Λ := { ≤ k ≤ τ : lm ( v k ) · lm ( g k ) = lm ( f ) , lc ( v k ) · lc ( g k ) ∈ R ∗ q } . Then we can obtain an identity of leading terms asfollows. lt ( f ) = λ − X k ∈ Λ lt ( v k ) · lt ( g k ) ∈ h lt ( B q ) i (6.33)since we have lc ( f ) ∈ R ∗ q as well as e q = 0 in (6.22) in this case.To obtain(6.32), for every f ∈ I p , let ι p be the injection defined as in (6.30)such that ι p ( f ) ∈ I q . Now consider the identity (6.22) that holds for ι p ( f ). We canapply the epimorphism σ p in (6.29) to the identity (6.22) for ι p ( f ) to obtain anidentity of leading terms that is similar to (6.33) since lc ( f ) ∈ R ∗ p and σ p ( e q ) = 0.The proofs for the equivalence between the characterizations in (1) and (2), aswell as between those in (3) and (4), are verbatim repetitions of that for Theorem6.13. In fact, it suffices to substitute I q , B q and σ q , as well as I p , B p and σ p , for I d , B d and σ d respectively. Remark . We used gcd -reductions in Theorem 6.13 and Theorem 6.15 (2) and(4) to address the ideal membership problem for the new type of bases. However weavoided making gcd -reductions of the S -polynomials in the computations of thepseudo-bases and pseudo-eliminants in Algorithm 3.9, as well as the proper basesand proper eliminants in Algorithm 5.20. The reason becomes clear in Section 8when we show that the gcd -computations not only incur complexity issues butalso contain B´ezout coefficients that tend to swell to an excruciating magnitudeover the rational field K = Q . This is also the reason why we do not adopt theso-called strong Gr¨obner bases for polynomial rings over principal ideal rings. Andthe PQR is a special kind of principal ideal rings. Please refer to [AL94, P251,Definition 4.5.6] for the definition of strong Gr¨obner bases over PIDs.50 otation . Let us use B q to denote the various modular bases that were defined previouslyas follows: (i) The proper basis B q with the moduli being a composite divisor q asin Definition 5.22; (ii) The modular basis B d = σ d ( B ε ) with the moduli being thecompatible part d = cp ( χ ε ) as in Lemma 6.6; (iii) The modular basis B p = σ p ( B q )with the moduli being a proper eliminant e q as in Theorem 6.15 (3).In accordance with the above notation of B q , we also use σ q as a unified notationfor the epimorphisms σ q in (5.30), σ d in (6.14) and σ p in (6.29). Moreover, wedenote the coefficient ring R q , R d or R p simply as R q and ideal I q , I d or I p as I q respectively such that B q ⊂ I q ⊂ R q [ ˜ x ] henceforth.The unified modular basis B q satisfies the similar identities (6.15), (6.31) and(6.32) that can be assimilated into a unified identity as follows. h lt ( I q ) i = h lt ( B q ) i . (6.34)Now we furnish the unified identity (6.34) with an interpretation via the ideal I + h q i . Let B ε = { g k : 1 ≤ k ≤ s } be a pseudo-basis of I and B q = σ q ( B ε ) = { b j : 1 ≤ j ≤ t } ⊂ I q the unified notation for the modular bases as in Notation6.17. We can also define a unified notation for the injection ι q : R q [ ˜ x ] → ( K [ x ])[ ˜ x ]similar to (5.31) and (6.30) such that σ q ◦ ι q is the identity map. By the canonicalisomorphism as follows:( I + h q i ) / h q i ≃ I/ ( I ∩ h q i ) ≃ σ q ( I ) = I q , it is easy to deduce that ι q ( B q ) ⊂ B ε + h q i := { g k + f q : 1 ≤ k ≤ s, f ∈ ( K [ x ])[ ˜ x ] } .Then it readily follows that ( ι q ( B q ) ∪ { } ) + h q i = B ε + h q i . Here ι q ( B q ) ∪ { } isfor the possibility that 0 ∈ σ q ( B ε ).We can deduce that for every f ∈ I + h q i , there exists g ∈ ( K [ x ])[ ˜ x ] suchthat f − gq ∈ h ι q ( B q ) i . In fact, we can invoke Theorem 6.13 and Theorem 6.15(2) and (4) on σ q ( f ) ∈ I q . According to the gcd -reduction by the modularbasis B q = { b j : 1 ≤ j ≤ t } as in (6.20), there exist a multiplier λ ∈ R × q andquotients h j ∈ R q [ ˜ x ] with 1 ≤ j ≤ t such that λ · σ q ( f ) = P tj =1 h j b j . Since h j b j = σ q ( ι q ( h j ) · ι q ( b j )) for 1 ≤ j ≤ t , there exists h ∈ ( K [ x ])[ ˜ x ] such that λ · ι q ( σ q ( f )) − hq = P tj =1 ι q ( h j ) · ι q ( b j ) ∈ h ι q ( B q ) i . Moreover, we also have f − ι q ( σ q ( f )) ∈ h q i . Hence f − gq ∈ h ι q ( B q ) i for some g ∈ ( K [ x ])[ ˜ x ].Thus we have proved the following interpretation for the identity (6.34) since( ι q ( B q ) ∪ { } ) + h q i = B ε + h q i . Lemma 6.18. If a modular basis B q satisfies the identity (6.34) , then B ε ∪ { q } or ι q ( B q ) ∪ { q } constitute a basis for I + h q i . Let B ε be a pseudo-basis and d = cp ( χ ε ) the compatible part of a pseudo-eliminant χ ε of I . We still use θ q to denote the nontrivial proper divisors for q ∈ Θwith Θ being the corresponding set of composite divisors as in (6.2). Then we havethe following new type of bases in accordance with the ideal decomposition of I in (6.2): ( B ε ∪ { d } ) ∪ [ q ∈ Θ ( B ∪ { θ q } ) , (6.35)where the basis B stands for either B ε or ι q ( B q ).51he modular version of the new type of bases specified in (6.35) correspond tothe modular bases B q , B d and B p as in Notation 6.17. These modular bases areespecially suited for the Chinese Remainder Theorem. Lemma 6.19. Let Θ be the set of composite divisors whose proper divisors arenontrivial as in (6.2) and (6.35) . We use the unified notation I q as in Notation6.17 for the modular ideals I q in (6.31) and I p in (6.32) but we keep the notationfor I d as in (6.15) unaltered. If we denote I χ := I/I ∩ h χ i , then we have adecomposition as follows. I χ ≃ I d × Y q ∈ Θ I q . (6.36) Proof. The identity (6.36) amounts to proving that the canonical homomorphism ϕ as follows is an isomorphism: ϕ : I/I ∩ h χ i −→ ( I/I ∩ h d i ) × Y q ∈ Θ I/I ∩ h q i . (6.37)The proof is similar to that for the Chinese Remainder Theorem. In fact, it isobvious that ϕ is an injection since we have the decomposition:( I ∩ h d i ) ∩ \ q ∈ Θ ( I ∩ h q i ) = I ∩ h χ i (6.38)by the factorization of the eliminant χ in Theorem 6.2.For every q ∈ Θ ∪ { d } := b Θ, let us define J q := T q ′ ∈ b Θ \{ q } ( I ∩ h q ′ i ). Fix a q ∈ b Θ.For every q ′ ∈ b Θ \ { q } , there exist c q , c q ′ ∈ K [ x ] such that c q q + c q ′ q ′ = 1. Hencefor every f ∈ I , we have the following identity: f = f Y q ′ ∈ b Θ \{ q } ( c q q + c q ′ q ′ ) ∈ ( I ∩ h q i ) + J q , from which we can deduce the following ideal identity for every q ∈ b Θ: I = ( I ∩ h q i ) + J q . (6.39)We can substitute (6.39) into (6.37) to obtain the following equivalence for every q ∈ b Θ: I/I ∩ h q i = (( I ∩ h q i ) + J q ) / ( I ∩ h q i ) ≃ J q / (( I ∩ h q i ) ∩ J q ) = J q / ( I ∩ h χ i ) , (6.40)where the last equality is based on (6.38). The identity (6.40) simplifies thecanonical monomorphism ϕ in (6.37) into the following form: ϕ : I/I ∩ h χ i −→ Y q ∈ b Θ J q / ( I ∩ h χ i ) ≃ Y q ∈ b Θ I/I ∩ h q i π q −→ I/I ∩ h q i , (6.41)where π q is the canonical projection onto the components. The surjectivity of ϕ follows from the fact that for s q ∈ J q with q ∈ b Θ, we have π q ◦ ϕ ( s ) = s q with s := P q ∈ b Θ s q . 52 emma 6.20. For the eliminant χ of an ideal I , let R χ := ( K [ x ])[ ˜ x ] / h χ i and I χ be defined as in (6.36) respectively. Then we have the following isomorphism inaccordance with the ideal decompositions in (6.2) and (6.38) : R χ /I χ ≃ ( R d /I d ) × Y q ∈ Θ ( R q /I q ) , (6.42) where Θ is defined as in (6.2) and (6.36) and I d and I q are defined as in (6.36) .Proof. By Chinese Remainder Theorem we have the algebra decomposition: R/ ( I + h χ i ) ≃ R/ ( I + h d i ) × Y q ∈ Θ R/ ( I + h q i ) (6.43)with R = ( K [ x ])[ ˜ x ]. Then the decomposition (6.42) follows from (6.43) by thefollowing observation: R/ ( I + h q i ) ≃ ( R/ h q i ) (cid:14) (( I + h q i ) / h q i ) ≃ R q /I q . Similarly we have R/ ( I + h χ i ) ≃ R χ /I χ and R/ ( I + h d i ) ≃ R d /I d .It is easy to show that the gcd -reduced remainder r obtained in the gcd -reduction (6.20) of Theorem 6.12 is unique. In this way we can define a unique normal form r for every f ∈ R q [ ˜ x ] with respect to B . This combined with theidentity (6.42) yield a normal form for every f ∈ R χ with respect to I χ .It is evident that Definition 6.9 and Lemma 6.10 apply to all these modularbases denoted as B q . In the remaining part of this section, let us address theuniqueness of this new type of bases. The first step is to minimize the number ofelements in a basis. Definition 6.21 (Irredundant basis) . A basis B q as in Notation 6.17 satisfying (6.34) is said to be irredundant if h lt ( B q \ { b } ) i is a proper subset of h lt ( B q ) i for every element b ∈ B q . That is, lt ( b ) is gcd -reduced with respect to B q \ { b } for every b ∈ B q by Lemma 6.10. Lemma 6.22. If B q and B ′ q are irredundant bases as in Definition 6.21 of the sameideal I q , then we have two equal sets of leading monomials lm ( B q ) = lm ( B ′ q ) .Proof. From lt ( b ) ∈ h lt ( B ′ q ) i for every b ∈ B q , we know that lt ( b ) is gcd -reducible with respect to B ′ q as per Lemma 6.10. Further, for every b ′ ∈ B ′ q | lm ( b ) asin Notation 6.8, lt ( b ′ ) is also gcd -reducible with respect to B q . Nonetheless weknow for sure that not every element of lt ( B ′ q | lm ( b ) ) be gcd -reducible with respectto B q \ { b } . Otherwise lt ( b ) would be gcd -reducible with respect to B q \ { b } ,contradicting the assumption that B q is irredundant. As a result, there exists a b ′ ∈ B ′ q | lm ( b ) satisfying b ∈ B q | lm ( b ′ ) . These two conditions indicate that lm ( b ) isdivisible by lm ( b ′ ) and vice versa. Hence we have lm ( b ′ ) = lm ( b ), from which wecan deduce that lm ( B q ) ⊂ lm ( B ′ q ) since b ∈ B q is arbitrary. Similarly we have lm ( B ′ q ) ⊂ lm ( B q ). Remark . The two irredundant bases B q and B ′ q in Lemma 6.22 do not haveto contain the same number of elements. In fact, consider the scenario of b j ∈ B q with j = 1 , lm ( b ) = lm ( b ) = min { lm ( B q ) } . Suppose that lc ( b j ) ∈ ∗ q \ R × q and lc ( b j ) /d ∈ R ∗ q \ R × q for j = 1 , d := gcd( lc ( b ) , lc ( b )). ByLemma 6.7 there exist c j ∈ R q for j = 1 , d = c · lc ( b ) + c · lc ( b ).Now we construct a new irredundant basis B ′ q by substituting b := c b + c b forboth b and b in B q . Then we still have h lt ( B q ) i = h lt ( B ′ q ) i as in (6.34) andmoreover, lm ( B q ) = lm ( B ′ q ). Definition 6.24 (Minimal basis) . An irredundant basis B q as in Definition 6.21 is called a minimal basis if forevery f ∈ B q , its leading coefficient lc ( f ) = λ · gcd( { lc ( b ) : b ∈ B q | lm ( f ) } ) with λ ∈ B × q being a unit. Here B q | lm ( f ) and gcd are defined as in Notation 6.8.In the example in Remark 6.23, the basis B q is irredundant but not minimal. Lemma 6.25. Let B q ⊂ R q [ ˜ x ] be an irredundant basis as in Definition 6.21. Forevery f ∈ B q , let us denote d := gcd( { lc ( b ) : b ∈ B q | lm ( f ) } as in Notation 6.8.Assume that d = P b ∈ B q | lm ( f ) c b · lc ( b ) with c b ∈ R q as in Lemma 6.7. For every f ∈ B q , if we substitute f by g := X b ∈ B q | lm ( f ) c b · lm ( f ) lm ( b ) b, (6.44) we shall obtain a new basis denoted as B ′ q . We delete every g ∈ B ′ q for which thereexists g ′ ∈ B ′ q with g ′ = g such that lt ( g ′ ) = λ · lt ( g ) with λ ∈ R × q . If we stilldenote the new basis set as B ′ q , then B ′ q is a minimal basis.Proof. We prove the irredundancy of B ′ q by contradiction. Assume that thereexists g ∈ B ′ q such that g is gcd -reducible with respect to B ′ q \ { g } . Let us denoteΩ := ( B ′ q \ { g } ) | lm ( g ) = ∅ as in Notation 6.8. That is, for every b ∈ Ω there is c b ∈ R q such that the following identity holds. lc ( g ) = X b ∈ Ω c b · lc ( b ) . (6.45)Moreover, for every b ∈ Ω, we have lm ( b ) ≺ lm ( g ). The reason is that if lm ( b ) = lm ( g ), we would have lt ( b ) = λ · lt ( g ) with λ ∈ R × q by the definition of B ′ q basedon (6.44). Then one of b and g would have been deleted from B ′ q .Now every b ∈ Ω satisfies lt ( b ) ∈ h lt ( B q ) i since Ω ⊂ B ′ q ⊂ I q and B q satisfiesthe identity h lt ( I q ) i = h lt ( B q ) i . Hence lt ( b ) is gcd -reducible with respect to B q as per Lemma 6.10, i.e., for every a ∈ B q | lm ( b ) := Γ b = ∅ , there exists h a ∈ R q such that lc ( b ) = P a ∈ Γ b h a · lc ( a ). This combined with (6.45) lead to: lc ( g ) = X b ∈ Ω X a ∈ Γ b c b h a · lc ( a ) . (6.46)From (6.46) we can infer that lt ( g ) is gcd -reducible with respect to S b ∈ Ω Γ b .Moreover, the construction of g ∈ B ′ q in (6.44) is based on an f ∈ B q such that lm ( f ) = lm ( g ) and lc ( f ) is divisible by lc ( g ). Hence lt ( f ) is gcd -reduciblewith respect to S b ∈ Ω Γ b . But we already proved that lm ( b ) ≺ lm ( g ) = lm ( f )for every b ∈ Ω. This indicates that f / ∈ S b ∈ Ω Γ b and thus we have S b ∈ Ω Γ b ⊂ B q | lm ( g ) \ { f } ⊂ B q \ { f } . Hence lt ( f ) is gcd -reducible with respect to B q \ { f } .This contradicts the assumption that B q is irredundant.The minimality of B ′ q readily follows from Definition 6.24.54 emma 6.26. If B q and B ′ q are minimal bases of the same ideal I q as in Definition6.24, then we have two equal sets of leading terms lt ( B q ) = lt ( B ′ q ) and moreover, B q and B ′ q have the same number of basis elements.Proof. Since minimal bases are irredundant, we already have lm ( B q ) = lm ( B ′ q ) asper Lemma 6.22. For every b ∈ B q , there exists b ′ ∈ B ′ q such that lm ( b ) = lm ( b ′ ).Moreover, lt ( b ′ ) is gcd -reducible with respect to B q and hence lc ( a ) is divisibleby lc ( b ′ ) for every a ∈ B q | lm ( b ′ ) due to the minimality of B ′ q ∋ b ′ . Now b ∈ B q | lm ( b ′ ) and hence lc ( b ) is divisible by lc ( b ′ ). Similarly lc ( b ′ ) is divisible by lc ( b ) as well.Thus we also have lc ( b ) = λ · lc ( b ′ ) with λ ∈ R × q .The conclusion for the number of elements is immediate from lt ( B q ) = lt ( B ′ q )and the irredundancy of B q and B ′ q . Definition 6.27 (Reduced basis) . A minimal basis B q as in Definition 6.24 is said to be a reduced basis if every b ∈ B q is gcd -reduced with respect to B q \ { b } . That is, every nonzero term of b is gcd -reduced with respect to B q \ { b } . Lemma 6.28. If both B q and B ′ q are reduced bases of the same ideal I q as inDefinition 6.27, then we have two equal bases B q = B ′ q .Proof. According to Lemma 6.26, we have lt ( B q ) = lt ( B ′ q ) since both of themare minimal bases. Hence a term c α ˜ x α is gcd -reduced with respect to B q if andonly if it is gcd -reduced with respect to B ′ q .Now for every b ∈ B q , there is b ′ ∈ B ′ q such that lt ( b ) = lt ( b ′ ). Let us assumethat b − b ′ = 0. By b − b ′ ∈ I q , we have lt ( b − b ′ ) ∈ h lt ( B q ) i as per the identity(6.34). Then we have lt ( b − b ′ ) ∈ h lt ( B q \ { b } ) i since lt ( b − b ′ ) ≺ lt ( b ). Hence lt ( b − b ′ ) is gcd -reducible with respect to B q \ { b } by Lemma 6.10.On the other hand, every nonzero term of b and b ′ is gcd -reduced with respectto B q \ { b } and B ′ q \ { b ′ } respectively by Definition 6.27. By lt ( B q \ { b } ) = lt ( B ′ q \ { b ′ } ), we can infer that every nonzero term of b ′ is gcd -reduced withrespect to B q \ { b } as well. Thus we can conclude that every nonzero term of b − b ′ is gcd -reduced with respect to B q \ { b } . In particular, lt ( b − b ′ ) is gcd -reducedwith respect to B q \ { b } . This constitutes a contradiction. Remark . Please note that a reduced basis is not necessarily a strong basislike the strong Gr¨obner basis. A strong Gr¨obner basis is defined as a finite set G := { g j : 1 ≤ j ≤ s } such that for every f ∈ h G i , there exists a g j ∈ G such that lt ( f ) is divisible by lt ( g j ). Consider the following counterexample.Let I = h f, g i ⊂ R q [ x, y ] be an ideal with basis f = ( z + 1) x + r ( z ) and g =( z − y + s ( z ) such that r, s ∈ R q . An invocation of Algorithm 5.20 showsthat their S -polynomial as in (5.16) can be properly reduced to 0. Hence { f, g } constitutes a proper basis of I by Definition 5.22 and satisfies the identity (6.31)by Theorem 6.15. Then it is easy to verify that { f, g } constitutes a reduced basisby Definition 6.27. Nevertheless it does not constitute a strong basis of I . Infact, consider h := yf − xg ∈ I . Then lt ( h ) = 2( z + 1) xy is divisible by neither lt ( f ) = ( z + 1) x nor lt ( g ) = ( z − y . Please also refer to [AL94, P251, Definition 4.5.6]. emark . If you are enticed by the uniqueness of the reduced ideal bases asin Lemma 6.28, you should be prepared to embrace the phenomenal sizes of theintermediate B´ezout coefficients such as the c b ’s in (6.44). We shall elaborate onthis issue when we make a complexity comparison between the new type of basesand Gr¨obner bases in Section 8. The exemplary wild growth of B´ezout coefficientscan be found especially in Example 8.1. We already strove to simplify our algorithms via Corollary 3.7 and Corollary 5.14as well as the triangular identities in Lemma 3.8 and Lemma 5.18. In this sectionwe make further improvements on the efficiency and complexity of Algorithm 3.9and Algorithm 5.20.Recall that in Algorithm 3.9 and Algorithm 5.20 we have temporary sets G and F respectively containing the basis elements, as well as the temporary sets S containing the S -polynomials. We also have Procedure P for the pseudo-reductionand proper reduction of the S -polynomials in S respectively. Let us supplementthe following principles to improve the efficiency of Algorithm 3.9 and Algorithm5.20. Principle . (i) We always list the elements in the temporary set G in Algorithm 3.9 andtemporary set F in Algorithm 5.20 such that their leading terms are in increasingorder with respect to the monomial ordering.(ii) When we make a pseudo-reduction or proper reduction of an S -polynomialin S in Procedure P , we always use the basis elements in the temporary set G in Algorithm 3.9 or F in Algorithm 5.20 first whose leading terms are as small aspossible with respect to the monomial ordering.(iii) When we choose an S -polynomial in S for pseudo-reduction or properreduction by Procedure P , we always choose the one whose leading term is assmall as possible with respect to the monomial ordering.(iv) For each triplet we can invoke a triangular identity as in (3.9) or (5.28) atmost once. Moreover, we choose the triangular identity such that the multiplier λ in (3.9) or (5.28) is as simple as possible. More specifically, in the case of (3.9)we require that the degree of the squarefree part of λ as in Definition 4.1 be assmall as possible. In the case of (5.28), we require that the degrees of both theunit factor λ × and standard factor λ st as in (5.6) be as small as possible. And thedegree of λ st has the priority for the comparison.(v) In Algorithm 5.20 when we initialize the temporary set F := σ q ( G ) byapplying the epimorphism σ q in (5.30) to the original basis G , we should choose therepresentations in R q of the coefficients of the basis elements in F such that theirsquarefree parts are as simple as possible in terms of both degrees and coefficients.The priority of the comparison is given to the degrees of the squarefree parts ofthe leading coefficients. This is especially true for the case when we already havethe factorizations of the coefficients.Principle 7.1 enhances efficiency by imposing a preference or direction for theimplementation of Algorithm 3.9 to follow. In effect whenever we make a pseudo-reduction of an S -polynomial in S , we usually obtain a remainder r with the least56eading term in G . Then the remainder r generates a new S -polynomial with theleast leading term in S . We choose to make a pseudo-reduction of this new S -polynomial in S according to Principle 7.1 (iii). A repetition of this process withstrictly decreasing leading terms inevitably leads to a temporary pseudo-eliminantin K [ x ] before we make a pseudo-reduction of all the S -polynomials in S . Let usdenote the temporary pseudo-eliminant as χ δ . It is easy to see that the pseudo-eliminant χ ε , the final output of Algorithm 3.9, satisfies χ δ ∈ ( χ ε ) ⊂ K [ x ].In what follows let us use the temporary pseudo-eliminant χ δ to further improveLemma 3.8 and Corollary 3.7. Lemma 7.2. Let χ δ be a temporary pseudo-eliminant in Algorithm 3.9 such that χ δ ∈ ( χ ε ) ⊂ K [ x ] with χ ε being the pseudo-eliminant. For f, g, h ∈ ( K [ x ])[ ˜ x ] \ K [ x ] , suppose that lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i . If the multiplier λ = lc ( h ) /d as in (3.9) is relatively prime to χ δ , then it is unnecessary to add λ into themultiplier set Λ in Procedure Q of Algorithm 3.9. We simply disregard the S -polynomial S ( f, g ) .Proof. The multiplier λ in (3.9) is relatively prime to the temporary pseudo-eliminant χ δ and hence the pseudo-eliminant χ ε as well as the eliminant χ . Theproof of Theorem 4.10 demonstrates that such kind of multipliers have impacton the soundness of neither our arguments nor our conclusions since they wouldappear as factors of the multiplier ν in (4.25). Lemma 7.3. Let χ δ be a temporary pseudo-eliminant in Algorithm 3.9 such that χ δ ∈ ( χ ε ) ⊂ K [ x ] with χ ε being the pseudo-eliminant. Suppose that lm ( f ) and lm ( g ) are relatively prime for f, g ∈ ( K [ x ])[ ˜ x ] \ K [ x ] . If d = gcd( lc ( f ) , lc ( g )) as in (3.5) is relatively prime to χ δ , then it is unnecessary to add λ = d intothe multiplier set Λ in Procedure Q of Algorithm 3.9. We simply disregard the S -polynomial S ( f, g ) . The reason for disregarding the multiplier d in Lemma 7.3 is the same as thatfor disregarding the multiplier λ in Lemma 7.2. Remark . Please note that after we obtain the temporary pseudo-eliminant χ δ in Algorithm 3.9, we refrain from simplifying the leading coefficients of the basiselements by substituting gcd( lc ( f ) , χ δ ) for lc ( f ) notwithstanding the temptationof converting f into a monic polynomial. The reason is that such simplificationsinvolve the B´ezout coefficients that most probably have gigantic sizes over Q .Please refer to Example 8.1 for an example on this.The conformity to Principle 7.1 during the implementation of Algorithm 5.20also yields a temporary proper eliminant denoted as e δ in most cases before theoutput of the proper eliminant e q such that e δ ∈ ( e q ) ⊂ R q . When e δ ∈ R ∗ q , similarto Lemma 7.2 and Lemma 7.3, we can make improvements on Lemma 5.18 andCorollary 5.14 as follows. Lemma 7.5. Let e δ ∈ R ∗ q be a temporary proper eliminant in Algorithm 5.20 suchthat e δ ∈ ( e q ) ⊂ R q with e q being the proper eliminant. For f, g, h ∈ ( R q [ ˜ x ]) ∗ \ R × q with at most one of them in R ∗ q \ R × q , suppose that lcm( lm ( f ) , lm ( g )) ∈ h lm ( h ) i . Ifthe multiplier λ = σ q ( l h /d ) as in (5.28) is relatively prime to the temporary propereliminant e δ , then we simply disregard the S -polynomial S ( f, g ) in Procedure R ofAlgorithm 5.20. 57e omit the proof of Lemma 7.5 since it is almost a verbatim repetition ofthat for Lemma 7.2. In fact, if the multiplier λ is relatively prime to the tempo-rary proper eliminant e δ , then so is it to the proper eliminant e q . Such kind ofmultipliers have impact on neither our arguments nor our conclusions in Theorem5.26. Lemma 7.6. Let e δ ∈ R ∗ q be a temporary proper eliminant in Algorithm 5.20 suchthat e δ ∈ ( e q ) ⊂ R q with e q being the proper eliminant. Suppose that lm ( f ) and lm ( g ) are relatively prime for f, g ∈ R q [ ˜ x ] \ R q . If d := gcd q ( lc ( f ) , lc ( g )) asin (5.20) is relatively prime to the temporary proper eliminant e δ , then we simplydisregard their S -polynomial S ( f, g ) in Procedure R of Algorithm 5.20.Remark . After we obtain a temporary proper eliminant e δ in Algorithm 5.20,we can simplify computations by implementing the remaining part of the algorithmin R p [ ˜ x ] over the normal PQR R p ≃ R q / ( e δ ), which are defined similar to (6.28)and (6.29).In Lemma 7.6 we can deduce that the multiplier d for the proper reduction isrelatively prime to the proper eliminant e q since e δ ∈ ( e q ) ⊂ R q . By Lemma 5.3(i) we have a unit multiplier σ p ( d ) ∈ R × p with σ p defined as in (6.29). Hence the S -polynomial S ( f, g ) can be disregarded for our conclusions. A conspicuous phenomenon in the computation of Gr¨obner bases over Q in lex ordering is the explosion of intermediate coefficients. The consumption of time andmemory in the computation of S -polynomials constitutes another burdensomecomputational complexity. In this section we prove two lemmas as exemplaryillustrations showing that our new type of bases minimize these two problems toa substantial extent.Let K be a field and f, g ∈ ( K [ x ]) ∗ . The polynomials u, v ∈ K [ x ] satisfying uf + vg = gcd( f, g ) are called the B´ezout coefficients of f and g .The following Example 8.1 shows that albeit f, g ∈ Q [ x ] have moderate integralcoefficients, their B´ezout coefficients can swell to quite unpalatable sizes. Example . f ( x ) := (7 x − x − x + 13 x + 29 x − x − x − x + 3 x + 1) ; g ( x ) := (6 x + 15 x + x − x − x + 64 x + 18 x + 5 x − x − x − . I refrain from printing out the B´ezout coefficients in Example 8.1 since theycan trigger a bit discomfort and be calculated for private appreciations via anypopular software for symbolic computations.For a field K and polynomial f ∈ K [ x ], let us use lt( f ), lm( f ) ∈ [ x ] andlc( f ) ∈ K ∗ to denote the leading term, leading monomial and leading coefficient of f over the field K respectively. This is to discriminate from our previous notationsfor the leading term lt ( f ), leading monomial lm ( f ) ∈ [ ˜ x ] and leading coefficient lc ( f ) ∈ K [ x ] of f over the PID K [ x ] when we treat K [ x ] as ( K [ x ])[ ˜ x ].58or f, g ∈ K [ x ] \ { } with lm( f ) ∈ h lm( g ) i , recall that after the first step ofpolynomial division of f by g , we have: h = f − lt( f )lt( g ) g. (8.1)A continuation of the polynomial division in (8.1) yields a representation: f = qg + r (8.2)with the quotient q and remainder r in K [ x ] such that r is reduced with respectto g , that is, r / ∈ h lm( g ) i \ { } .Also recall that in the computation of Gr¨obner bases over the field K , the S -polynomial of f, g ∈ K [ x ] \ { } over K is defined as: S ( f, g ) := x γ lt( f ) f − x γ lt( g ) g (8.3)with x γ := lcm(lm( f ) , lm( g )) ∈ [ x ] .Please note that when lm( f ) ∈ h lm( g ) i in (8.3), then x γ = lm( f ) and wehave the following relationship between the S -polynomial S ( f, g ) in (8.3) andpolynomial division in (8.1): lc( f ) · S ( f, g ) = h. (8.4) Lemma 8.2. For a field K and elimination ordering z ≺ x on K [ x, z ] , considerthe ideal I := h ax + c, bx + d i with a, b, c, d ∈ ( K [ z ]) ∗ . Suppose that I is a zero-dimensional ideal such that ad − bc = 0 . Then the computation of Gr¨obner basisof I contains Euclidean algorithm for the computation of gcd( a, b ) . In particular,the B´ezout coefficients of a and b appear in the intermediate coefficients of thecomputation.Proof. Without loss of generality, suppose that lt( a ) = c α z α and lt( b ) = c β z β with c α , c β ∈ K ∗ and α ≥ β .Suppose that the first step of polynomial division of ax + c by bx + d as in (8.1)is a x + c := ax + c − ( c α /c β ) z α − β ( bx + d ). According to the identity in (8.4), theclassical S -polynomial as in (8.3) is essentially a x + c up to a unit multiplier c α ,that is, c α S ( ax + c, bx + d ) = a x + c . If we have deg( a ) ≥ deg( b ), a continuationof the division of a x + c by bx + d as in (8.2) coincides with the reduction of the S -polynomial c α S ( ax + c, bx + d ) by bx + d in the computation of Gr¨obner basisfor I : c α S ( ax + c, bx + d ) = a x + c = q ( bx + d ) + b x + d (8.5)with q, b , d ∈ K [ z ] such that deg( b ) < deg( b ).Let us assume that b d = 0 in (8.5). In the computation of Gr¨obner basis, weadd b x + d into the temporary set of basis for I in this case. Then we computethe S -polynomial S ( bx + d, b x + d ) and further reduce it by b x + d . Similarto the above discussion in (8.5) based on the identity (8.4), this exactly coincideswith the step of Euclidean algorithm in which we make a polynomial division of bx + d by b x + d .A repetition of the above process shows that the computation of Gr¨obner basisfor I amounts to an implementation of Euclidean algorithm for the computation of59cd( a, b ), i.e., the greatest common divisor of the leading coefficients a, b ∈ ( K [ z ]) ∗ ,albeit with unit multipliers like c α = lc( a ) in (8.5). Let us denote ρ := gcd( a, b )with B´ezout coefficients u, v ∈ K [ z ] such that ρ = ua + vb . According to Euclideanalgorithm, after the above computations we shall obtain u ( ax + c ) + v ( bx + d ) = ρx + uc + vd . We can obtain the same result via the computation and reduction of S -polynomials if we assimilate the unit multipliers like c α in (8.5) into the B´ezoutcoefficients u, v . We add ρx + uc + vd into the temporary set of basis for I anduse it to eliminate the variable x so as to obtain the eliminant χ of I . Let usdenote a = mρ and b = nρ with m, n ∈ K [ z ]. Similar to the above discussions,the process of reducing the S -polynomial c α S ( ax + c, ρx + uc + vd ) by ρx + uc + vd amounts to the elimination of the variable x as follows. ax + c − m ( ρx + uc + vd ) = c − m ( uc + vd ) = cρ − a ( uc + vd ) ρ = v ( bc − ad ) ρ (8.6)since ρ = ua + vb . Similarly we have: bx + d − n ( ρx + uc + vd ) = − u ( bc − ad ) ρ . (8.7)From u ( a/ρ ) + v ( b/ρ ) = 1 we can infer that the B´ezout coefficients u and v arerelatively prime to each other. Hence the eliminant χ can be obtained from (8.6)and (8.7) as following: χ = gcd (cid:16) v ( bc − ad ) ρ , − u ( bc − ad ) ρ (cid:17) = bc − adρ . (8.8)The B´ezout coefficients u and v appear in (8.6) and (8.7) but not in the eliminant χ in (8.8).Lemma 8.2 represents a more generic scenario than the ideal I = h ax + c, bx + d i appears to be. In fact, in the final steps in the computation of the eliminant ofa zero-dimensional ideal, we often run into the situation in the lemma. In thecase of Lemma 8.2 over the PID K [ z ], the intermediate coefficients in (8.6) and(8.7) contain the B´ezout coefficients u and v of the leading coefficients a and b as their factors that tend to swell over the rational field Q like in Example 8.1.Nonetheless in terms of our new type of S -polynomials as in (3.1) over the PID K [ z ], we have a straightforward computation as follows. S ( ax + c, bx + d ) = λ ( ax + c ) − µ ( bx + d ) = λc − µd = bc − adρ = χ, (8.9)where the two multipliers λ := l/a = b/ρ = n and µ := l/b = a/ρ = m with l := lcm( a, b ) and ρ = gcd( a, b ) such that a = mρ and b = nρ .The eliminant χ in (8.8) is obtained in one step in (8.9) without resorting tothe B´ezout coefficients u and v of the leading coefficients a and b .Now let us generalize Lemma 8.2 to contrive a generic scenario as follows. Lemma 8.3. With a field K and elimination ordering on [ x ] as in Definition3.1, suppose that the generators f and g of the ideal I = h f, g i ⊂ ( K [ x ])[ ˜ x ] satisfy lt ( f ) = a ˜ x α and lt ( g ) = b ˜ x β with the leading coefficients a, b ∈ ( K [ x ]) ∗ .Then the computation of Gr¨obner basis of I contains Euclidean algorithm for thecomputation of gcd( a, b ) . In particular, the B´ezout coefficients of a and b appearin the intermediate coefficients of the computation. roof. Without loss of generality, suppose that s = deg( a ) ≥ deg( b ) = t . Letus denote lc( f ) = c and lc( g ) = d in K ∗ respectively. Then lt( f ) = cx s ˜ x α andlt( g ) = dx t ˜ x β . With ˜ x γ := lcm( ˜ x α , ˜ x β ), we have lcm(lm( f ) , lm( g )) = x s ˜ x γ . The S -polynomial in (8.3) now bears the following form: cS ( f, g ) = ˜ x γ − α f − cx s − t d ˜ x γ − β g = (cid:16) a − cx s − t d b (cid:17) ˜ x γ + ˜ x γ − α (cid:16) f − lt( f )lt( g ) g (cid:17) (8.10)with f := f − lt ( f ) = f − a ˜ x α and g := g − lt ( g ) = g − b ˜ x β .Since we also have c = lc( a ) and d = lc( b ) respectively, the leading coefficient of cS ( f, g ) in ( K [ x ])[ ˜ x ] in (8.10), i.e., lc ( cS ( f, g )) = a − ( c/d ) x s − t b := a ∈ K [ x ],is exactly the first step of the polynomial division of a by b in K [ x ].If deg( a ) ≥ deg( b ), then a further reduction of the S -polynomial cS ( f, g ) in(8.10) by g amounts to a polynomial division of a by b as in (8.2). Suppose thatwe have a = qb + r with q, r ∈ K [ x ] such that deg( r ) < deg( b ). Then cS ( f, g ) isreduced by g to a polynomial h ∈ ( K [ x ])[ ˜ x ] bearing the form h = r ˜ x γ + h suchthat lt ( h ) = r ˜ x γ . For simplicity, let us assume that h is already reduced withrespect to g , that is, no term of h is in h lm( g ) i ⊂ K [ x ].Next in the computation of Gr¨obner basis, we add h into the basis { f, g } of I and compute the S -polynomial S ( g, h ). Similar to (8.10), the leading coefficient lc ( dS ( g, h )) of ˜ x γ amounts to the first step of the polynomial division of b by r in K [ x ]. This together with a further reduction of dS ( g, h ) by h exactly coincidewith the step of Euclidean algorithm in which we make a polynomial division of b by r in K [ x ].A repetition of the above process shows that the computation of Gr¨obner basisfor I contains an implementation of Euclidean algorithm for the computation ofgcd( a, b ), i.e., the greatest common divisor of the leading coefficients a = lc ( f )and b = lc ( g ) in K [ x ], albeit with unit multipliers like c in (8.10). Let usdenote ρ := gcd( a, b ) with B´ezout coefficients u, v ∈ K [ x ] such that ρ = ua + vb .In essence the computation of Gr¨obner basis amounts to a computation of thegreatest common divisor of the leading coefficients. Hence based on Euclideanalgorithm we have: u ˜ x γ − α ( a ˜ x α + f ) + v ˜ x γ − β ( b ˜ x β + g ) = ρ ˜ x γ + u ˜ x γ − α f + v ˜ x γ − β g := w. (8.11)We add w in (8.11) into the basis of I and then compute the S -polynomial S ( f, w ). We make a reduction of the S -polynomial S ( f, w ) by w . Let us denote a = mρ and b = nρ with m, n ∈ K [ z ]. From the perspective of ( K [ x ])[ ˜ x ],this reduction process can be summarized as being equivalent to the followingelimination of the leading term a ˜ x α of f = a ˜ x α + f :˜ x γ − α f − mw = ρ [(1 − mu ) ˜ x γ − α f − mv ˜ x γ − β g ] /ρ = v ( b ˜ x γ − α f − a ˜ x γ − β g ) ρ . (8.12)Similarly we have: ˜ x γ − β g − nw = − u ( b ˜ x γ − α f − a ˜ x γ − β g ) ρ . (8.13)Thus the B´ezout coefficients u and v appear in the computation of Gr¨obnerbasis for I . And they might swell over the rational field K = Q .61ith l := lcm( a, b ), let us denote two multipliers λ := l/a = b/ρ = n and µ := l/b = a/ρ = m . In terms of the S -polynomials as in (3.1) over the PID K [ x ],we have a straightforward computation of the S -polynomial S ( f, g ) as follows. S ( f, g ) = λ ˜ x γ − α ( a ˜ x α + f ) − µ ˜ x γ − β ( b ˜ x β + g )= λ ˜ x γ − α f − µ ˜ x γ − β g = ( b ˜ x γ − α f − a ˜ x γ − β g ) /ρ. (8.14)We obtained a simpler result in (8.14) in one step than those in (8.12) and (8.13)without the B´ezout coefficients u and v of the leading coefficients a = lc ( f ) and b = lc ( g ) that might swell to an unexpected size over the rational field K = Q like in Example 8.1. I furnish this section with two examples to demonstrate the computations of thenew type of bases for zero-dimensional ideals. In theses examples it is conspicuousthat the intermediate coefficients as well as the coefficients of the basis elementsare restrained to moderate sizes and do not swell like in the case of Gr¨obner basesover Q .Example 9.1 is an excerpt from the textbook [CLO05, Chapter 8, P426, § χ ε procured in thisway is exactly the eliminant χ . Example . Suppose that the ideal I = h f, g, h i ⊂ Q [ x, y, z ] with f = − x + y + z − g = − zx + y + 2; h = x + x − zy. (9.1)For the purpose of comparison, we list its classical Gr¨obner basis with respectto the lex ordering z ≺ y ≺ x as { p, g , g } such that: p = z − z − z + 4 z + 6 z + 14 z − z − z + z + 9 z + 6; g = 38977 y + 1055 z + 515 z + 42 z − z − z + 5285 z −− z + 36881 z + 7905 z + 42265 z − z − g = 38977 x + 1055 z + 515 z + 42 z − z − z + 5285 z −− z + 36881 z + 7905 z + 3288 z − z + 1791 . (9.2)Let us denote the temporary pseudo-basis set G := { f, g, h } . As per Principle7.1 (i), we list the elements in G in increasing order of their leading terms withrespect to the lex ordering as in (9.1).As in Procedure Q of Algorithm 3.9, we can disregard the S -polynomial S ( g, h )as per the triangular identity (3.9). In fact, lcm( lt ( g ) , lt ( h )) = − zx is divisibleby lt ( f ) = − x and hence the multiplier λ = 1 in (3.9) in this case. The temporary S -polynomial set is S = { S ( f, g ) , S ( f, h ) } .According to Principle 7.1 (iii), we first compute the S -polynomial: S ( f, g ) = zf − g = − y + zy + z − z − e. (9.3)We add it into G such that G = { e, f, g, h } . We name it as the first element e according to Principle 7.1 (i) because lt ( e ) is less than every element in lt ( G \{ e } )62nd hence cannot be pseudo-reduced by G \{ e } as in Theorem 2.7. Then we delete S ( f, g ) from S .We can disregard the S -polynomials S ( e, f ), S ( e, g ) and S ( e, h ) as in Procedure Q of Algorithm 3.9. This is based on Corollary 3.7 since lt ( e ) is relatively primeto every element in lt ( G \ { e } ).We compute the S -polynomial S ( f, h ) = xf + h = xy + z x − zy and pseudo-reduce its leading term xy by f like in (2.1) as per Principle 7.1 (ii).The remainder of the term pseudo-reduction is as follows: r = S ( f, h ) + yf = z x + y + ( z − z − y. We make a pseudo-reduction of the leading term lt ( r ) = z x by f like in (2.1) foranother time. The final remainder is as follows. d := r + z f = y + (2 z − z − y + z ( z − 1) (9.4)that cannot be further pseudo-reduced by G . We add it into G such that G = { d, e, f, g, h } . The reason for naming it as the first element d of G is still Principle7.1 (i). Then we delete S ( f, h ) from S .We disregard the S -polynomials S ( d, f ), S ( d, g ) and S ( d, h ) as in Procedure Q of Algorithm 3.9 based on Corollary 3.7. We add S ( d, e ) into S such that S = { S ( d, e ) } .We compute the S -polynomial S ( d, e ) as following: S ( d, e ) = yd + e = (2 z − z − y + ( z − z + 1) zy + z − z − z − z − y by d as in (2.1). For the samereason as above, we name the remainder of the term pseudo-reduction as the firstelement in G such that: c := S ( d, e ) − (2 z − z − d = (3 z − z − z + z + 1) y + 2 z − z − z + z + z + 2 . (9.5)Now G = { c, d, e, f, g, h } . Then we delete S ( d, e ) from S .We disregard the S -polynomials S ( c, f ), S ( c, g ) and S ( c, h ) as in Procedure Q of Algorithm 3.9 based on Corollary 3.7 due to their relatively prime leading terms.By the triangular identity (3.9), we also disregard the S -polynomial S ( c, e ) as inProcedure Q of Algorithm 3.9 since lcm( lt ( c ) , lt ( e )) = − (3 z − z − z + z +1) y is divisible by lt ( d ) = y . Here the multiplier λ as in (3.9) satisfies λ = 1. Weadd S ( c, d ) into S such that S = { S ( c, d ) } .We compute the S -polynomial S ( c, d ) = − yc + (3 z − z − z + z + 1) d = y (4 z − z + 8 z + 2 z −− z − 3) + (3 z − z − z + 5 z + 3 z − z − z and then pseudo-reduce it by c . The multiplier for the pseudo-reduction is 3 z − z − z + z + 1 and we add it into a multiplier set Λ = { z − z − z + z + 1 } .The remainder of the pseudo-reduction is a temporary pseudo-eliminant χ δ := z − z − z + 4 z + 6 z + 14 z − z − z + z + 9 z + 6 . S ( c, d ) from S such that S = ∅ and we have completed theprocedure of Algorithm 3.9. Hence the pseudo-eliminant χ ε = χ δ . Moreover,the pseudo-eliminant χ ε is relatively prime to the multiplier λ in Λ and hence iscompatible. Hence the eliminant χ = χ ε according to Theorem 4.10.Now the pseudo-basis of I as in Definition 3.11 is B ε = G = { c, d, e, f, g, h } as in(9.1), (9.3), (9.4) and (9.5) respectively. Let us define q := χ and the epimorphism σ q : ( K [ z ])[ x, y ] −→ R q [ x, y ] as in (6.14) over the normal PQR R q ≃ K [ z ] / ( q ).Now both lt ( σ q ( g )) = − zx and lt ( σ q ( h )) = x are divisible and hence gcd -reducible with respect to σ q ( f ) as in Definition 6.9. Thus in order to obtain anirredundant modular basis as in Definition 6.21, we can delete σ q ( g ) and σ q ( h )from B q = σ q ( B ε ). Moreover, we also delete σ q ( d ) and σ q ( e ) from B q since both lt ( σ q ( d )) = y and lt ( σ q ( e )) = − y are gcd -reducible with respect to σ q ( c ) asper Definition 6.9 for gcd -reducibility. In fact, we have lc ( σ q ( c )) = 3 z − z − z + z + 1 ∈ R × q is a unit. Hence the following basis constitutes an irredundantbasis of I q = σ q ( I ) under the above epimorphism σ q : σ q ( c ) = (3 z − z − z + z + 1) y + 2 z − z − z + z + z + 2; σ q ( f ) = − x + y + z − . (9.6)For simplicity we also call { c, f } an irredundant basis of I .The irredundant basis in (9.6) is automatically a minimal basis as in Definition6.24 but not a reduced basis as in Definition 6.27. We can make a gcd -termreduction of the term y in σ q ( f ) by σ q ( c ) with multiplier µ = 3 z − z − z + z +1 ∈ R × q as in Definition 6.11 to obtain a remainder denoted as b . Now { σ q ( c ) , b } constitutes a reduced basis for I q . For simplicity we still denote σ q ( c ) as c andthey are our new type of basis for I q that we still denote as B q as follows. (cid:26) c = (3 z − z − z + z + 1) y + 2 z − z − z + z + z + 2; b = (3 z − z − z + z + 1) x − z + 3 z + 2 z − z − z + 2 z + 3 . (9.7)The reduced basis B q of I q in (9.7) is unique by Lemma 6.28. It satisfies theidentity (6.15) in Lemma 6.6 and hence the unified identity (6.34). Since b = ι q ( b )and c = ι q ( c ), the set { b, c, χ } constitutes another form of our new type of basis for I + h χ i = I . It is in a simpler form than the one in (9.2) with moderate coefficientsand exponents for the variable z .The following Example 9.2 is a slight complication of Example 9.1 in that themultiplier set Λ is not empty during the implementation of Algorithm 3.9. Hencewe have to invoke Algorithm 5.20 over a PQR with zero divisors to procure theexact form of the eliminant and new type of bases. The coefficients of the classicalGr¨obner basis in Example 9.2 also cause a bit more psychological disturbancesthan those in Example 9.1. Example . Suppose that the ideal I = h f, g, h i ⊂ Q [ x, y, z ] with f = − z ( z + 1) x + y ; g = z ( z + 1) x − y ; h = − x y + y + z ( z − . (9.8)For the purpose of comparison, in the following we list its classical Gr¨obnerbasis G = { p, g , g , g , g } with respect to the lex ordering z ≺ y ≺ x : p = ( z − z ( z + 9 z + 36 z + 84 z + 126 z + 126 z + 85 z ++ 31 z + 19 z − z + 4 z − z − z − . (9.9)64 = 20253807 z y + 264174124 z + 1185923612 z + 850814520 z −− z − z + 1862876196 z + 12815317453 z ++ 3550475421 z + 2124010584 z − z + 42918431554 z −− z + 35649844325 z − z + 3388659963 z ++ 930240431 z − z − z ; g = 20253807 y + 903303104 z + 4102316224 z + 3140448384 z −− z − z + 4804720290 z + 43739947868 z ++ 14906482335 z + 9051639768 z − z ++ 139970660534 z − z + 118589702914 z −− z + 11927452134 z + 2021069107 z − z −− z ; g = 2592487296 z x + (7777461888 z − y + 108083949263 z ++ 486376518055 z + 349557551130 z − z −− z + 788268739077 z + 5350420983851 z ++ 1476923019345 z + 689330555757 z − z ++ 17386123487861 z − z + 13787524468420 z −− z + 786997920594 z + 628350552934 z −− z − z ; g = 20253807 x y + 1037047036 z + 4686773132 z + 3455561112 z −− z − z + 6731446644 z + 51651585868 z ++ 16267315284 z + 7429467573 z − z ++ 163168836472 z − z + 135706468958 z −− z + 11263865469 z + 2500312823 z + 197272975 z −− z − z + 20253807 z . We define the temporary pseudo-basis set G := { f, g, h } as in (9.8). As perPrinciple 7.1 (i), we list the elements in G in increasing order of their leadingterms with respect to the lex ordering as in (9.8).We can disregard the S -polynomial S ( g, h ) as in Procedure Q of Algorithm3.9 based on the triangular identity in Lemma 3.8. In fact, lcm( lt ( g ) , lt ( h )) = − z ( z + 1) x y is divisible by lt ( f ) = − z ( z + 1) x and hence in this case themultiplier λ = 1 in (3.9). We shall not take into account the triangular identityof S ( f, h ) with respect to g since we already invoked a triangular identity in theabove on the same triplet { f, g, h } according to Principle 7.1 (iv). Hence thetemporary S -polynomial set is S = { S ( f, g ) , S ( f, h ) } .According to Principle 7.1 (iii), we first compute the S -polynomial S ( f, g ) = z ( z + 1) f + g = − y + z ( z + 1) y := e (9.10)that cannot be further pseudo-reduced by G as in Theorem 2.7. We add e into G such that G = { e, f, g, h } . We name it as the first element e in G because lt ( e ) isless than every element in lt ( G \ { e } ). Then we delete S ( f, g ) from S such that S = { S ( f, h ) } .We can disregard the S -polynomials S ( e, f ) and S ( e, g ) like in Procedure Q of Algorithm 3.9 based on Corollary 3.7 since lt ( e ) is relatively prime to lt ( f )65nd lt ( g ). We can also disregard the S -polynomial S ( e, h ) as in Procedure Q of Algorithm 3.9 based on the triangular identity (3.9) with respect to f . Infact, the leading monomials of the triplet satisfy lcm( lm ( e ) , lm ( h )) = x y beingdivisible by lm ( f ) = x . The multiplier λ in the identity (3.9) equals λ = lc ( f ) = − z ( z + 1) . We add λ into the multiplier set Λ such that Λ = { z ( z + 1) } .We compute the S -polynomial S ( f, h ) = z ( z + 1) h − xyf = − xy + z ( z + 1) y + z ( z + 1) ( z − and pseudo-reduce it as in Theorem 2.7 by e and f according to Principle 7.1 (ii).More specifically, we first pseudo-reduce lt ( S ( f, h )) = − xy by e in (9.10) withinterim multiplier µ = 1 as in (2.1) to obtain the following remainder: r = S ( f, h ) − xe = − z ( z + 1) xy + z ( z + 1) y + z ( z + 1) ( z − . Then we make a further pseudo-reduction of lt ( r ) = − z ( z + 1) xy by f in (9.8)as in (2.1) also with interim multiplier µ = 1. The new remainder is as follows. r = r − yf = z ( z + 1) y − y + z ( z + 1) ( z − . An ensuing pseudo-reduction of lt ( r ) = z ( z + 1) y by e in (9.10) with interimmultiplier µ = 1 yields the following remainder: r = r + z ( z + 1) ye = ( z ( z + 1) − y + z ( z + 1) ( z − . We obtain the following final remainder d after a repetition of the above pseudo-reduction of lt ( r ) = ( z ( z + 1) − y by e in (9.10) with interim multiplier µ = 1: d := r + ( z ( z + 1) − e = z ( z + 1) [( z ( z + 1) − y + z ( z − ] . (9.11)For the same reason as above, we name the remainder d as the first element in G such that G = { d, e, f, g, h } . Then we delete S ( f, h ) from S such that S = ∅ .The leading monomial lm ( d ) = y is relatively prime to lm ( f ) = lm ( g ) = x . But their leading coefficients satisfy gcd( lc ( d ) , lc ( f )) = gcd( lc ( d ) , lc ( g )) = z ( z +1) , which is already in the multiplier set Λ. Hence we can just disregard the S -polynomials S ( d, f ) and S ( d, g ) as in Procedure Q of Algorithm 3.9. Moreover,we also disregard the S -polynomial S ( d, h ) by the triangular identity with respectto f as in (3.9) since lcm( lt ( d ) , lt ( h )) = z ( z + 1) ( z ( z + 1) − x y is divisibleby lt ( f ) = − z ( z + 1) x . We add the S -polynomial S ( d, e ) into S such that S = { S ( d, e ) } .We compute the S -polynomial S ( d, e ) as following: S ( d, e ) = yd + z ( z + 1) ( z ( z + 1) − e = z ( z + 1) ( z + 9 z + 36 z + 84 z + 126 z + 126 z + 85 z ++ 31 z + 19 z − z + 4 z − z − z − y. Then we make a pseudo-reduction of lt ( S ( d, e )) = S ( d, e ) by d in (9.11) as in(2.1) with the interim multiplier µ = z ( z + 1) − 1. The remainder of the termpseudo-reduction is: r = z ( z + 1) ( z + 9 z + 36 z + 84 z + 126 z + 126 z + 85 z + 31 z ++ 19 z − z + 4 z − z − z − z ( z + 1) − y + z ( z − ] . 66e add the multiplier µ into the multiplier set Λ such thatΛ = { z ( z + 1) , z ( z + 1) − } . (9.12)After a further pseudo-reduction of the above remainder r by d in (9.11) withinterim multiplier µ = 1, we can obtain a temporary pseudo-eliminant as follows. χ δ = ( z − z ( z + 1) ( z + 9 z + 36 z + 84 z + 126 z + 126 z ++ 85 z + 31 z + 19 z − z + 4 z − z − z − . (9.13)Then we delete S ( d, e ) from S such that S = ∅ . Since we have exhausted all the S -polynomials in S , the pseudo-eliminant χ ε = χ δ .By a comparison as in Algorithm 4.6 between the pseudo-eliminant χ ε in (9.13)and multiplier set Λ in (9.12), we can compute the compatible part cp ( χ ε ) of thepseudo-eliminant χ , which is defined in Definition 4.5, as following: cp ( χ ε ) = ( z − ( z + 9 z + 36 z + 84 z + 126 z + 126 z + 85 z ++ 31 z + 19 z − z + 4 z − z − z − . (9.14)Moreover, the composite divisors of the incompatible part ip ( χ ε ) of the pseudo-eliminant χ ε in (9.13) are z and ( z + 1) as per Definition 4.7.For the composite divisor q = z , in what follows let us invoke Algorithm 5.20to compute its corresponding proper eliminant e q . The computations are basedon Theorem 5.26 over the normal PQR R q ≃ K [ z ] / ( z ).The epimorphism σ q as in (5.30) transforms the basis elements g and h into: σ q ( g ) = (20 z + 15 z + 6 z + 1) z x − y ; σ q ( h ) = − x y + y + (10 z − z + 5 z − z . The squarefree part of lc ( g ) in (9.8) equals z ( z + 1) whereas it equals z (20 z +15 z + 6 z + 1) as above. Hence we should use the representation of g in (9.8) asper Principle 7.1 (v). The same reason for using the representation of h in (9.8).Thus although we start with the basis elements F = { σ q ( f ) , σ q ( g ) , σ q ( h ) } , we stilldenote it as { f, g, h } in (9.8) henceforth with the understanding that F ⊂ R q [ x, y ]and R q ≃ K [ z ] / ( z ).We can disregard the S -polynomial S ( g, h ) as in Procedure R of Algorithm5.20 by the triangular identity with respect to f as in Lemma 5.18. In fact, bythe representations in (9.8), we have lcm( lt ( g ) , lt ( h )) = − z ( z + 1) x y beingdivisible by lt ( f ) = − z ( z + 1) x . Hence it is easy to deduce that the multiplier λ in (5.28) now becomes λ = 1 and thus it is redundant to compute the S -polynomials S ( g, h ). The temporary S -polynomial set is S = { S ( f, g ) , S ( f, h ) } .According to Principle 7.1 (iii), we first compute the S -polynomial S ( f, g ): S ( f, g ) = z ( z + 1) f + g mod ( q = z )= − y + z ( z + 1) y := e. (9.15)The S -polynomial S ( f, g ) cannot be properly reduced by F = { f, g, h } . We add S ( f, g ) into F and name it as the first element e in F according to Principle 7.1(i) such that F = { e, f, g, h } . Then we delete S ( f, g ) from S .67e can disregard the S -polynomials S ( e, f ) and S ( e, g ) as in Procedure R of Algorithm 5.20 based on Corollary 5.14 because lt ( e ) is relatively prime toboth lt ( f ) and lt ( g ). We add the S -polynomial S ( e, h ) into S such that S = { S ( e, h ) , S ( f, h ) } .By Principle 7.1 (iii) we first compute the S -polynomial S ( f, h ): S ( f, h ) = xyf − z ( z + 1) h mod ( q = z )= xy − z ( z + 1) y − (2 z − z . We make a proper reduction of S ( f, h ) as in Theorem 5.10 by e and f accordingto Principle 7.1 (ii). More specifically, we first properly reduce lt ( S ( f, h )) = xy by e in (9.15) with interim multiplier µ = 1 as in (5.12). The remainder r is asfollows. r = S ( f, h ) + xe = z [( z + 1) xy − ( z + 1) y + z − z ] . Then we make a further proper term reduction of lt ( r ) = z ( z + 1) xy by f alsowith interim multiplier µ = 1. The remainder of the reduction is as follows. r = r + yf = − z ( z + 1) y + y − z + z . An ensuing proper term reduction of lt ( r ) = − z ( z + 1) y by e with interimmultiplier µ = 1 yields the following remainder: r = r − z ( z + 1) ye = ( − z − z − z − z + 1) y − z + z . We obtain the following final remainder after a repetition of the above proper termreduction of lt ( r ) by e with interim multiplier µ = 1: d := z [( − z − z + z + 3 z + 3 z + 1) y − z + z ] . (9.16)According to Principle 7.1 (i), we name the remainder d as the first element in F such that F = { d, e, f, g, h } . Then we delete S ( f, h ) from S .We disregard the S -polynomials S ( d, g ) and S ( d, h ) like in Procedure R ofAlgorithm 5.20 by the triangular identities with respect to f as in Lemma 5.18.In fact, in the case of S ( d, g ) the multiplier λ as in (5.28) satisfies λ = 1 whereasin the case of S ( d, h ) the multiplier λ = ( z + 1) ∈ R × q . We add the S -polynomials S ( d, e ) and S ( d, f ) into S such that S = { S ( d, e ) , S ( d, f ) , S ( e, h ) } .By Principle 7.1 (iii) we compute the S -polynomial S ( d, e ) first: S ( d, e ) = yd + ( − z − z + z + 3 z + 3 z + 1) z e mod ( q = z )= z (18 z + 16 z + 6 z + 1) y. We make a proper term reduction of lt ( S ( d, e )) by d as in (5.12) with interimmultiplier µ = − z − z + z + 3 z + 3 z + 1 ∈ R × q . The remainder of the reductionis 0 over R q . We delete S ( d, e ) from S .By Principle 7.1 (iii) we then compute the S -polynomial S ( d, f ): S ( d, f ) = ( z + 1) xd + ( − z − z + z + 3 z + 3 z + 1) yf mod ( q = z )= ( z + 1) z x + ( − z − z + z + 3 z + 3 z + 1) y . 68e make a proper term reduction of lt ( S ( d, f )) = ( z + 1) z x by f in (9.8) withinterim multiplier µ = ( z + 1) ∈ R × q as in (5.12). The remainder of the termreduction is: r = − ( z + 1) (9 z + z − z − z − z − y + z y. Next we make a proper term reduction of lt ( r ) by e in (9.15) with interimmultiplier µ = 1 to obtain a new remainder: r = z (42 z + 69 z + 56 z + 29 z + 8 z + 1) y. A further proper term reduction of lt ( r ) by d in (9.16) with interim multiplier µ = − z − z + z + 3 z + 3 z + 1 ∈ R × q as in (5.12) yields a temporary propereliminant: e δ = − z (6 z + 1) . (9.17)Then we delete S ( d, f ) from S .Since e δ has a standard representation e st δ = z , according to Remark 7.7, wecan simplify computations by implementing the algorithm over the normal PQR R p ≃ R q / ( z ), which is similar to (6.29).Let us now compute the final S -polynomial S ( e, h ) ∈ S in R p [ x, y ] as follows. S ( e, h ) = x e − yh mod ( q = z )= z ( z + 1) x y − y + ( z − z ) y. This is followed by a proper term reduction of lt ( S ( e, h )) = z ( z + 1) x y by h as in (5.12) with interim multiplier µ = 1. The remainder of the term reductionis as follows. r = S ( e, h ) + z ( z + 1) h = − y + z ( z + 1) y + ( z − z ) y. A further proper term reduction of lt ( r ) = − y by e also with interim multiplier µ = 1 leads to the remainder r = r − y e = z (1 − z ) y . We make a properterm reduction of this remainder r by d as in (5.12) with interim multiplier µ = − z − z + z + 3 z + 3 z + 1 ∈ R × p . The remainder of the reduction is 0 over R p . We delete S ( e, h ) from S such that S = ∅ .For the Procedure Q in Algorithm 5.20, now we have e st δ = z . Hence in(5.19) the multiplier n f = z and the S -polynomial S ( f, e st δ ) = z y over R p . Aproper term reduction of S ( f, e st δ ) by d ∈ F in (9.16) with the multiplier − z − z + z + 3 z + 3 z + 1 ∈ R × p yields the remainder 0 over R p . We also disregardthe S -polynomial S ( g, e st δ ). In fact, over the normal PQR R q = K [ z ] / ( z ) asabove, we have lcm( l g , l e ) = z ( z + 1) is divisible by l f = − z ( z + 1) with l g := ι q ( lc ( g )) = z ( z + 1) , l e := ι q ( e st δ ) = z and l f := ι q ( lc ( f )) = − z ( z + 1) .Hence we can invoke the triangular identity (5.28) on S ( g, e st δ ) with respect to f to show that the S -polynomial S ( g, e st δ ) can be disregarded. Moreover, back to thenormal PQR R p = K [ z ] / ( z ), we can prove that the S -polynomial S ( d, e st δ ) = 0 asin (5.19).For the composite divisor q = ( z + 1) as in (9.13), our computations are overthe normal PQR R q ≃ K [ z ] / (( z + 1) ). Under the epimorphism σ q as in (5.30),the ideal I q := σ q ( I ) is generated by F := σ q ( G ) ⊂ R q [ x, y ] with G as in (9.8): f = y ; g = − y ; h = − x y + y + z ( z − . h in (9.8) by Principle 7.1(v). And we abuse the notations a bit and still use { f, g, h } to denote the elementsin F . It is easy to corroborate that F is not consistent such that the ideal I q = { } and should be disregarded.Now the pseudo-basis of I as in Definition 3.11 is B ε := G = { d, e, f, g, h } asin (9.8), (9.10) and (9.11). To obtain an irredundant basis of I q as in Definition6.21 over the normal PQR R q ≃ K [ z ] / ( q ) with q := cp ( χ ε ) as in (9.14), we delete g from B ε since lt ( g ) = z ( z + 1) x is divisible by lt ( f ) = − z ( z + 1) x . Infact, under the epimorphism σ q : ( K [ z ])[ x, y ] → R q [ x, y ] as in (6.14), lt ( σ q ( g ))is gcd -reducible with respect to σ q ( f ) as per Definition 6.9 for gcd -reducibility.Similarly we also delete h from B ε since lt ( σ q ( h )) = − x y is gcd -reducible withrespect to σ q ( f ) based on lc ( σ q ( f )) = − z ( z + 1) ∈ R × q . Further, we delete e in(9.10) from B ε since lt ( σ q ( e )) = − y is gcd -reducible with respect to σ q ( d ) as in(9.11) due to the fact that lc ( σ q ( d )) = ( z + 1) z ( z ( z + 1) − ∈ R × q . Hence anirredundant basis of I q , which we denote as B q , is B q = { σ q ( d ) , σ q ( f ) } with d and f being defined in (9.11) and (9.8) respectively.This irredundant basis B q is automatically a minimal basis as in Definition6.24 but not a reduced basis as in Definition 6.27. We can make a proper termreduction of the term y in f by d as in (5.12) to obtain a remainder denoted as c . In this way we obtain a reduced basis that we still denote as B q . That is, B q = { c, σ q ( d ) } with d defined in (9.11) and c := − z ( z + 1) ( z ( z + 1) − x − z ( z + 1) ( z − . (9.18)The reduced basis B q = { c, σ q ( d ) } is unique by Lemma 6.28 and satisfies theidentity (6.15) in Lemma 6.6 and hence the unified identity (6.34).For the composite divisor q = z , our computations over the normal PQR R q ≃ K [ z ] / ( z ) started with the basis F of I q that bears the same form as the onein (9.8) and ended with the proper eliminant e q = z as the standard factor of e δ in (9.17). Hence let us consider the modular basis of I p = σ p ( I ) over the normalPQR R p ≃ R q / ( z ). What we already have is a modular basis F = { d, e, f, g, h } of I q = σ q ( I ) over R q ≃ K [ z ] / ( z ) that are defined in (9.8), (9.15) and (9.16)respectively.The basis elements g and h of I q in (9.8) would bear the following form over R p : g = z (6 z + 1) x − y ; h = − x y + y + (5 z − z . Nonetheless by Principle 7.1 (v), we still use the representations of g and h in(9.8). In order to have an irredundant proper basis of I p , we delete σ p ( g ) from σ p ( F ) since lt ( σ p ( g )) = z ( z + 1) x is divisible by lt ( σ p ( f )) = − z ( z + 1) x .The supplemented basis element e in (9.15) is invariant under the epimorphism σ p whereas the supplemented basis element d in (9.16) bears the form σ p ( d ) = z ( z + 1) y over R p . Now it is evident that we can use σ p ( d ) to make a termreduction of σ p ( e ) such that it bears a reduced form denoted as b := y . Moreover,we can render σ p ( h ) reduced by b such that σ p ( h ) = − x y + z ( z − .Altogether we obtain a reduced basis of I p denoted as B p over R p ≃ K [ z ] / ( z )as follows. B p (cid:26) b := σ p ( d ) = z ( z + 1) y ; b := − σ p ( f ) = z ( z + 1) x − y ; b = y ; b := − σ p ( h ) = x y − z ( z − . (9.19)70ith the compatible part q = cp ( χ ε ) of the pseudo-eliminant χ ε defined in(9.14), a reduced basis B q of I q is defined in (9.18) and (9.11) respectively asfollows. B q (cid:26) a := σ q ( d ) = z ( z + 1) [( z ( z + 1) − y + z ( z − ]; a := c = z ( z + 1) [( z + 1) ( z ( z + 1) − x + z ( z − ] . (9.20)Let ι q be the injection as in (5.31) associated with R q ≃ K [ z ] / ( z ). In thisexample we have a unique proper divisor θ q = ι q ( e st δ ) = z and hence the properfactor χ ip = θ q = z as per Definition 6.1. By Theorem 6.2 the eliminant χ = cp ( χ ε ) · χ ip with cp ( χ ε ) as in (9.14). This coincides with the eliminant p obtainedin the classical Gr¨obner basis as in (9.9). Nonetheless our new type of bases in(9.19) and (9.20) not only have much more moderate coefficients than those of theclassical Gr¨obner basis under (9.9) but also completely obviate the intermediatecoefficient swell problem.Moreover, based on the modular bases B q in (9.20) and B p in (9.19), we canuse Lemma 6.18 to obtain the new type of bases ι q ( B q ) ∪ { q } and ι p ( B p ) ∪ { z } for I + h q i and I + h z i respectively. Here q = cp ( χ ε ) is as in (9.14) and ι p as in(6.30). According to Lemma 6.3, we have I = ( I + h q i ) ∩ ( I + h z i ). 10 Conclusion and Remarks In this paper we defined a new type of bases as in (6.34) and (6.35) in accordancewith a decomposition of the original ideal in (6.36) and (6.2) respectively. Thecharacterizations of the new type of bases in Theorem 6.13 and Theorem 6.15 arein effect solutions to the ideal membership problem. The computations and logicaldeductions in this paper suggest that it is much easier to study a zero-dimensionalideal over principal quotient rings (PQR) modulo the factors of its eliminant oreven pseudo-eliminant than over fields.An obvious direction for future research is to generalize this new type of basesto ideals of positive dimensions. The new type of bases and their algorithms canbe easily generalized to such kind of ideals I ⊂ Z [ x ] as I ∩ Z = { } . We shalladdress the generic case of I ∩ Z = { } in forthcoming papers. It is meaningfulto enhance the computational efficiency of the new type of bases by the normalselection strategies and signatures as well as the conversions between differentmonomial orderings as aforementioned in the Introduction. A complexity analysison the new type of bases that is similar to those in [MM82] [MM84] [May89][Dub90] [KM96] [MR13] on Gr¨obner bases should be interesting. References [AL94] Adams W. and Loustaunau P., An Introductin to Gr¨obner Bases. Grad.Stud. Math. 3., Amer. Math. Soc. (1994)[Arn03] Arnold E., Modular algorithms for computing Gr¨obner bases. J. SymbolicComput. 35., P403-419 (2003)[BW93] Becker T. and Weispfenning V., Gr¨obner Bases. A Computational Ap-proach to Commutative Algebra. Grad. Texts in Math. 141., Springer (1993)71BW98] Buchberger B. and Winkler F., Gr¨obner Bases and Applications. LondonMath. Soc. Lecture Note Ser. 251., Cambridge Univ. Press (1998)[Buc65] Buchberger B., 1965 Ph.D. Thesis: An algorithm for finding the basiselements of the residue class ring of a zero dimensional polynomial ideal. J.Symbolic Comput. 41(3-4)., P475-511 (2006)[Buc85] Buchberger B., Gr¨obner Bases: An algorithmic method in polynomialideal theory, in Multidimensional Systems Theory, P184-232, ed. by Bose N.,D. Reidel Publishing (1985)[CKM97] Collart S., Kalkbrener M. and Mall D., Converting bases with theGr¨obner walk. J. Symbolic Comput. 24., P465-469 (1997)[CLO15] Cox D., Little J. and O’Shea D., Ideals, Varieties, and Algorithms. AnIntroduction to Computational Algebraic Geometry and Commutative Alge-bra. 4th Ed. Springer-Verlag (2015)[CLO05] Cox D., Little J. and O’Shea D., Using Algebraic Geometry. 2nd Ed.Grad. Texts in Math. 185., Springer-Verlag (2005)[Dub90] Dub´e T., The structure of polynomial ideals and Gr¨obner bases. SIAMJ. Comput. 19(4)., P750-773 (1990)[DL06] Decker W. and Lossen C., Computing in Algebraic Geometry. Springer-Verlag (2006)[EF17] Eder C. and Faug`ere J., A survey on signature-based algorithms for com-puting Gr¨obner bases. J. Symbolic Comput. 80., P719-784 (2017)[EH19] Eder C. and Hofmann T., Efficient Gr¨obner bases computation over prin-cipal ideal rings. J. Symbolic Comput. (2019), https://doi.org/10.1016/j.jsc.2019.10.020[EH12] Ene V. and Herzog J., Gr¨obner Bases in Commutative Algebra. Grad.Stud. Math. 130., Amer. Math. Soc. (2012)[Ebe83] Ebert, G. Some comments on the modular approach to Gr¨obner-bases.ACM SIGSAM Bulletin 17., P28-32 (1983)[FGL93] Faug`ere J., Gianni P., Lazard D. and Mora T., Efficient computation ofzero-dimensional Gr¨obner bases by change of ordering. J. Symbolic Comput.16., P329-344 (1993)[Fau02] Faug`ere J., A new efficient algorithm for computing Gr¨obner bases with-out reduction to zero (F5), in ISSAC 2002, Proceedings of the 2002 Interna-tional Symposium on Symbolic and Algebraic Computation, P75-83, ACMPress (2002)[Fro97] Fr¨oberg R., An Introduction to Gr¨obner Bases. John Wiley & Sons (1997)[GG13] von zur Gathen J. and Gerhard J., Modern Computer Algebra. 3rd Ed.Cambridge Univ. Press (2013) 72GMN91] Giovini A., Mora T., Niesi G., Robbiano L. and Traverso C., “Onesugar cube, please,” or selection strategies in the buchberger algorithm, inISSAC 1991, Proceedings of the 1991 International Symposium on Symbolicand Algebraic Computation, P49-54, ed. by Watt S., ACM Press (1991)[GP08] Greuel G. and Pfister G., A Singular Introduction to Commutative Alge-bra. Springer-Verlag (2008)[Gra93] Gr¨abe H., On Lucky Primes. J. Symbolic Comput. 15., P199-209 (1993)[KM96] K¨uhnle K. and Mayr E., Exponential space computation of Gr¨obner bases,in ISSAC 1996, Proceedings of the 1996 International Symposium on Sym-bolic and Algebraic Computation, P63-71, ACM Press (1996)[KR00] Kreuzer M. and Robbiano L., Computational Commutative Algbera 1.Springer-Verlag (2000)[MM82] Mayr E. and Meyer A., The complexity of the word problems for commu-tative semigroups and polynomial ideals. Adv. Math. 46(3)., P305-329 (1982)[MM84] M¨oller H. and Mora F., Upper and lower bounds for the degree of Gr¨obnerbases. EUROSAM 84. Lecture Notes in Comput. Sci. 174, P172-183 (1984).[MR13] Mayr E. and Ritscher S., Dimension-dependent bounds for Gr¨obner basesof polynomial ideals. J. Symbolic Comput. 49., P78-94 (2013)[May89] Mayr E., Membership in polynomial ideals over Q is exponential spacecomplete. in Proceedings of the 6th Annual Symposium on Theoretical As-pects of Computer Science, STACS’89, ed. by Monien B. and Cori R., LectureNotes in Comput. Sci. 349., P400-406, Springer-Verlag (1989)[Mol88] M¨oller H., On the construction of Gr¨obner bases using syzygies. J. Sym-bolic Comput. 6(2), P345-359 (1988)[Pau92] Pauer F., On lucky ideals for Gr¨obner basis computations. J. SymbolicComput. 14., P471-482 (1992)[Pau07] Pauer F., Gr¨obner bases with coefficients in rings. J. Symbolic Comput.42(11)., P1003-1011 (2007)[ST89] Sasaki T. and Takeshima T., A modular method for Gr¨obner-basis con-struction over Q and solving system of algebraic equations. J. InformationProcessing 12., P371-379 (1989).[Stu95] Sturmfels B., Gr¨obner Bases and Convex Polytopes. Univ. Lecture Ser.8., Amer. Math. Soc. (1995)[Tra89] Traverso C., Gr¨obner Trace Algorithms, in Symbolic and Algebraic Com-putations (Rome 1988), Lecture Notes in Comput. Sci. 358., P125-138,Springer-Verlag (1989)[Win87] Winkler F., A pp