Computing strong regular characteristic pairs with Groebner bases
aa r X i v : . [ c s . S C ] J u l Computing strong regular characteristic pairswith Gr ¨obner bases ✩ Rina Dong a, ∗ , Dongming Wang a,b a Beijing Advanced Innovation Center for Big Data and Brain Computing – LMIB – School of Mathematical Sciences,Beihang University, Beijing 100191, China b Centre National de la Recherche Scientifique, 75794 Paris cedex 16, France
Abstract
The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the re-duced lexicographical Gr¨obner basis of the ideal. A pair ( G , C ) of polynomial sets is a strong reg-ular characteristic pair if G is a reduced lexicographical Gr¨obner basis, C is the W-characteristicset of the ideal hGi , the saturated ideal sat( C ) of C is equal to hGi , and C is regular. In this paper,we show that for any polynomial ideal I with given generators one can either detect that I isunit, or construct a strong regular characteristic pair ( G , C ) by computing Gr¨obner bases suchthat I ⊆ sat( C ) = hGi and sat( C ) divides I , so the ideal I can be split into the saturated idealsat( C ) and the quotient ideal I : sat( C ). Based on this strategy of splitting by means of quotientand with Gr¨obner basis and ideal computations, we devise a simple algorithm to decompose anarbitrary polynomial set F into finitely many strong regular characteristic pairs, from which tworepresentations for the zeros of F are obtained: one in terms of strong regular Gr¨obner basesand the other in terms of regular triangular sets. We present some properties about strong regularcharacteristic pairs and characteristic decomposition and illustrate the proposed algorithm andits performance by examples and experimental results. Keywords:
Strong regular, characteristic decomposition, W-characteristic set, Gr¨obner basis,ideal computation
1. Introduction
Triangular sets [28, 34, 17, 31] and Gr¨obner bases [6, 7, 5, 10] are special kinds of well-structured sets of multivariate polynomials that can be used to represent and to study zeros ofarbitrary polynomial sets and ideals. A large variety of problems in commutative algebra andalgebraic geometry [10, 11] may readily be solved by transforming the involved sets of polyno-mials into triangular sets or Gr¨obner bases. It is widely known that the theories and methodsof triangular sets are di ff erent from those of Gr¨obner bases conceptually and operationally. Thequestions that have motivated our work here and in [32, 33] are what inherent relationship there ✩ This work was partially supported by the National Natural Science Foundation of China (NSFC 11771034). ∗ Corresponding author: [email protected], School of Mathematical Sciences, Beihang University, Beijing100191, China.
Email addresses: [email protected] (Rina Dong), [email protected] (Dongming Wang) ay exist between triangular sets and Gr¨obner bases and how to connect or combine the twoalgorithmic approaches to amplify their applicability and power. These questions have beentouched in [18, 23, 12, 20] for bivariate, zero-dimensional, and other kinds of special polynomialideals. For general polynomial ideals of arbitrary dimension, it is shown in [2, 32] that intrinsicalconnections between Ritt characteristic sets and lexicographical Gr¨obner bases exist and theconcept of W-characteristic sets plays an essential role in exploiting such connections. Apartfrom the lack of investigation on the connection aspect, the literature on triangular sets andGr¨obner bases is extremely rich (see [2, 5, 7, 4, 8, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 26,27, 29, 30, 32, 33] and references therein).Multivariate polynomials in a triangular set may be ordered strictly according to their leadingvariables, with respect to a fixed variable ordering, so the number of the polynomials cannot bebigger than that of the variables in any triangular set. On the other hand, Gr¨obner bases aredefined with respect to a fixed term order determined by the variable ordering and the number ofelements in a Gr¨obner basis can be arbitrarily large. For any polynomial ideal with given set F of generators, one can compute, by using any of the available algorithms, a Gr¨obner basis thatgenerates the same ideal as F . To represent the zeros of F using triangular sets, in general oneneeds more than one triangular set, so decomposition takes place. When a triangular set T is ofconcern, the leading coe ffi cients of the polynomials in T with respect to their leading variables,called the initials of the polynomials in T , play a fundamental role. The saturated ideal of T bythe product of the initials of the polynomials in T is simply called the saturated ideal of T anddenoted as sat( T ). Its radical is the largest ideal whose zero set contains the set of those zeros of T which are not zeros of any of the initials.Triangular sets may be ordered according to the ranks (leading variables and degrees) andthen the leading terms of their polynomials. Let the minimal triangular set contained in thereduced lexicographical (lex) Gr¨obner basis of a polynomial ideal (or trivially [1] if the ideal isunit) be called the W-characteristic set of the ideal. By strong regular Gr¨obner basis, we mean areduced lex Gr¨obner basis G such that the W-characteristic set C of the ideal hGi is regular andsat( C ) = hGi ; we call ( G , C ) a strong regular characteristic pair, or an src pair for short. Thepair ( G , C ) is an interesting object of study because the strong regular Gr¨obner basis G and theregular triangular set C therein not only have remarkable properties but also provide two di ff erentyet correlated representations for the zeros of the ideal hGi .What interests us most is algorithmic decomposition of arbitrary polynomial sets into strongregular Gr¨obner bases, or equivalently into src pairs, for which we have the following generalapproach. From any polynomial set F , one can compute finitely many regular sets T , . . . , T e (also called regular chains; [16]) such that p hF i = p sat( T ) ∩ · · · ∩ p sat( T e ) . (1)There are two families of algorithms for such regular triangular decomposition. One family ofalgorithms was proposed initially by Kalkbrener [17], and developed further by Moreno Mazaand coauthors [25, 1, 8, 9] with algorithmic techniques from the method of Lazard [19]. Thesealgorithms were designed mainly for computing regular triangular representations of the form(1). The other family of algorithms was proposed by the second author [31, 30] to computeregular zero decompositions of the form Z ( F ) = Z ( T / J ) ∪ · · · ∪ Z ( T e / J e ) , (2)where each J i is the product of the initials of the polynomials in the regular set T i and Z ( T i / J i )denotes the set of all common zeros of the polynomials in T i which do not make J i vanish. It is2asy to see that decomposition (2) implies representation (1). However, representation (1) doesnot necessarily lead to decomposition (2) and the generators of the saturated ideals sat( T i ) in(1) are not explicitly provided. Nevertheless, for each sat( T i ) a Gr¨obner basis can be computedstraightforwardly from T i .Another alternative approach proposed recently by Mou and the authors [33] permits oneto decompose an arbitrary polynomial set into normal or regular characteristic pairs directly.This approach is based on a structure theorem about irregular W-characteristic sets [32] forthe splitting of ideals and relies strongly upon Gr¨obner basis computation; it is independentof pseudo-division-based triangular decomposition.In this paper, we show that from any polynomial ideal I with given generators one can eitherdetect that I is unit, or construct an src pair ( G , C ) such that I ⊆ sat( C ) and sat( C ) divides I bycomputing Gr¨obner bases. After ( G , C ) (called an src divisor of I ) is constructed, one can dividethe saturated ideal sat( C ) out of I to obtain an ideal J by taking ideal quotient: J = I : sat( C ).Then the radical ideal of I is decomposed as the intersection of the radical ideals of sat( C ) and J . Two key techniques are used in search for an src divisor of I : one is to produce an src pair bymeans of computing sat(wcs( · · · sat(wcs( I )))), where wcs( I ) denotes the W-characteristic set of I . The other is to use the relation sat( C ∗ ) , I , when the produced src pair is not an src divisor of I , to determine a polynomial F ∈ sat( C ∗ ) \ I with C ∗ = wcs( I ). In this case, there exists a positiveinteger q such that H = F [ Q C ∈C ∗ ini( C )] q ∈ hC ∗ i ⊆ I and an src divisor of I may be found amongthe src pairs computed from h I ∪ { H i }i for H i < I and H i | H . Once J is obtained, one can iteratethe find-and-divide process with J instead of I . Based on this strategy of splitting by meansof quotient and with Gr¨obner basis and ideal computations, we devise a simple algorithm todecompose an arbitrary polynomial set F into finitely many src pairs ( G , C ) , . . . , ( G e , C e ) suchthat p hF i = p sat( C ) ∩ · · · ∩ p sat( C e ) = p hG i ∩ · · · ∩ p hG e i . (3)In the above decomposition computed by the ideal-division-based algorithm, each G i is ac-tually the reduced lex Gr¨obner basis of a saturated ideal sat( T i ), where T i is the W-characteristicset of a certain ideal that contains hF i . The regular set C i is the W-characteristic set of sat( T i ),obtained from G i as a by-product for free. Moreover, regular sets computed by our algorithm arenormal in most cases and they are often simpler than the corresponding regular sets computedby pseudo-division or subresultant-based algorithms, because the former are minimal triangularsets taken from lex Gr¨obner bases. In general, the generating sets of the saturated ideals of normaltriangular sets are much easier to compute than those of abnormal ones. More importantly, ouralgorithm generates few redundant components, so the decomposition process usually terminatesin a few iterations. The e ff ectiveness of the decomposition algorithm has been demonstrated byour experimental results.The rest of the paper is organized as follows. After a brief introduction to Gr¨obner bases,triangular decomposition, and characteristic decomposition in Section 2, we show in Section 3how to construct from an arbitrary ideal I an src pair ( G , C ) such that I ⊆ sat( C ) = hGi . InSection 4, it is explained how an src divisor of I can be produced from any ideal I ( K [ x ] andthe ideal-division-based algorithm for strong regular characteristic decomposition is described. Most of the available methods for triangular decomposition use pseudo-division to eliminate variables. Indoing pseudo-division, the dividend has to be multiplied by some power of the initial of the dividing polynomial.Repeated multiplication of the powers of initials necessarily creates extraneous factors, making the sizes of intermediatepolynomials increase rapidly. It happens often that decomposition cannot continue after a few successive pseudo-divisions because polynomials in the pseudo-reminder sequence become larger and larger.
2. Preliminaries
In this section we recall some basic notions which will be used in the following sections.For those notions which are not formally introduced in the paper, the reader may consult thereferences [2, 31, 5, 10].
Let K be a field of characteristic 0 and K [ x , . . . , x n ] be the ring of polynomials in n orderedvariables x < · · · < x n with coe ffi cients in K . Throughout the paper, we write x for ( x , . . . , x n ).Let F be a polynomial in K [ x ] \ K . With respect to the variable ordering, the greatest variableappearing in F is called the leading variable of F and denoted as lv( F ). Assume that lv( F ) = x i ; then F can be written as F = I x ki + R , where I ∈ K [ x , . . . , x i − ], R ∈ K [ x , . . . , x i ], anddeg( R , x i ) < k = deg( F , x i ) (the degree of F in x i ). The polynomial I is called the initial of F ,denoted as ini( F ). For any polynomial set F ⊆ K [ x ], ini( F ) stands for { ini( F ) : F ∈ F } . Definition 1.
A finite, nonempty, ordered set [ T , . . . , T r ] of polynomials in K [ x ] \ K is called a triangular set if lv( T ) < · · · < lv( T r ).We denote by prem( P , Q ) the pseudo-reminder of P ∈ K [ x ] with respect to Q ∈ K [ x ] \ K inlv( Q ). Let T = [ T , . . . , T r ] ⊆ K [ x ] be any triangular set; the pseudo-reminder of P with respectto T is defined as prem( P , T ) : = prem( · · · prem(prem( P , T r ) , T r − ) , . . . , T ) . The variables in { x , . . . , x n } \ { lv( T ) , . . . , lv( T r ) } are called the parameters of T . For any twopolynomial sets F , G ⊂ K [ x ], define Z ( F / G ) : = { ¯ x ∈ ¯ K n : F ( ¯ x ) = , G ( ¯ x ) , , for all F ∈ F , G ∈ G} , where ¯ K is the algebraic closure of K . Sometimes we write Z ( F / Q G ∈G G ) for Z ( F / G ) and write Z ( F ) for Z ( F / ∅ ).Let F ⊆ K [ x ] be any polynomial set and denote by hF i the ideal generated by F in K [ x ]and by √hF i the radical of hF i . For any P ⊆ K [ x ], hF i : hPi denotes the ideal quotient of hF i by hPi . The saturated ideal of a triangular set T = [ T , . . . , T r ] is defined as sat( T ) : = { P ∈ K [ x ] : ∃ i such that PJ i ∈ hT i} , where J = ini( T ) · · · ini( T r ). For any c ∈ K \ { } , we consider[ c ] also as a triangular set, which is trivial, and define sat([ c ]) = h i . Definition 2.
Let T = [ T , . . . , T r ] be any nontrivial triangular set in K [ x ]. T is said to be regular , or called a regular set or a regular chain , if ini( T i ) is neither zero nor a zero-divisor in K [ x ] / sat( T i − ) for all i = , . . . , r .Regular sets or chains [30, 2] are special triangular sets with nice properties which have beenextensively studied. In particular, it is proved in [2, 31] that a triangular set T is regular if andonly if sat( T ) = { P ∈ K [ x ] : prem( P , T ) = } . The triangular set T is called a normal set (orsaid to be normal ) if ini( T ) does not involve any of the leading variables of the polynomials in T . Obviously, any normal set is regular, while a regular set is not necessarily normal.4or a given term order < , the greatest term in a polynomial F ∈ K [ x ] with respect to < iscalled the leading term of F and denoted as lt( F ). In this paper, we are concerned only with < lex ,the lex term order. Definition 3.
Let I ⊆ K [ x ] be an ideal, < be a term order, and h lt( I ) i stand for the ideal generatedby the leading terms of all the polynomials in I . A finite set { G , . . . , G s } ⊆ I is called a Gr¨obnerbasis of I with respect to < if h lt( G ) , . . . , lt( G s ) i = h lt( I ) i .Let G = { G , . . . , G s } be a Gr¨obner basis of an ideal I ⊆ K [ x ] with respect to a fixed termorder < . For any polynomial F ∈ K [ x ], there exists a unique polynomial R ∈ K [ x ], called the normal form of F with respect to G and denoted as nform( F , G ), such that F − R ∈ I and noterm of R is divisible by any of lt( G ) , . . . , lt( G s ). If F = R , then F is said to be B-reduced withrespect to G . Definition 4.
A Gr¨obner basis { G , . . . , G s } is said to be reduced if every G i is monic and noterm of G i is divisible by any lt( G j ) for all j , i and i , j = , . . . , s .In the rest of this paper, the variable ordering will be fixed and all Gr¨obner bases mentionedare meant reduced lex Gr¨obner bases. For any polynomial ideal, one can compute its unique reduced lex Gr¨obner basis and fromthe Gr¨obner basis, one can extract a minimal triangular set. This special triangular set, definedformally as the W-characteristic set of the ideal, possesses remarkable properties and plays a keyrole in our work on src pairs.
Definition 5 ([32, Def. 3.1]) . Let F be a polynomial set in K [ x ], G be the Gr¨obner basis of hF i , Θ = G ∩ K , G h i i = { G ∈ G : lv( G ) = x i } , Θ i be the set consisting of the smallest polynomial in G h i i if G h i i , ∅ , or ∅ otherwise (1 ≤ i ≤ n ). The set Θ ∪ Θ ∪ · · · ∪ Θ n of polynomials, orderedwith increasing leading variables when G , { } , is called the W-characteristic set of hF i . Proposition 6.
Let I be any ideal in K [ x ] with I (0) = I ∩ K = ∅ , I ( i ) = I ∩ K [ x , . . . , x i ] , I h i i = { F ∈ I ( i ) \ I ( i − : F is monic and B-reduced with respect to all P ∈ I ( i − } , and Ω i be theset consisting of the smallest polynomial in I h i i with respect to the lex term order if I h i i , ∅ , or ∅ otherwise (1 ≤ i ≤ n ) . Then the set Ω ∪ · · · ∪ Ω n of polynomials, ordered with increasing leadingvariables, is the W-characteristic set of I .Proof. Let G be the reduced lexicographical Gr¨obner basis of I . We need to show that Ω ∪· · · ∪ Ω n defined in the proposition is the minimal triangular set contained in G . Note that G h i i = G ( i ) \ G ( i − ⊆ I h i i for 1 ≤ i ≤ n , where G (0) = ∅ and G ( i ) = G ∩ K [ x , . . . , x i ]. For each i , if Ω i = ∅ , then I h i i = ∅ , so G h i i = ∅ .Now suppose that Ω i , ∅ for some i and let Ω i = { F i } . Then F i is B-reduced with respect toall the polynomials in I ( i − ⊇ G ( i − . If G h i i = ∅ , then F i , G ( i ) .This leads to a contradiction because F i ∈ I ( i ) . Hence G h i i , ∅ . Let G i be the smallest polynomialin G h i i . Then lt( F i ) ≤ lt( G i ) with respect to the lexicographical term order. If lt( F i ) < lt( G i ), then F i is B-reduced with respect to G ( i ) . This contradicts with the fact that F i ∈ I ( i ) . Thereforelt( F i ) = lt( G i ) > lt( F i − G i ), for F i and G i are monic. Since F i is the smallest polynomial in I h i i , F i − G i ∈ I ( i − . As F i and G i are both B-reduced with respect to G ( i − , so is F i − G i . It followsthat F i − G i ≡
0, and thus F i = G i . 5he ordered set Ω ∪ · · · ∪ Ω n in the above proposition equals the W-characteristic set of I , so it can be taken as an equivalent definition for the W-characteristic set of any nontrivialpolynomial ideal. This alternative definition, which is independent of Gr¨obner basis and isworked out in response to a comment of Mingsheng Wang, does not indicate how to constructthe W-characteristic set of the ideal. Proposition 7 ([32, Prop. 3.1]) . Let F be a polynomial set in K [ x ] and C be the W-characteristicset of hF i ⊆ K [ x ] . Then: ( a ) for any F ∈ hF i , prem( F , C ) = ; ( b ) hCi ⊆ hF i ⊆ sat( C ) ; ( c ) Z ( C / ini( C )) ⊆ Z ( F ) ⊆ Z ( C ) . We say that the variable ordering condition is satisfied for a triangular set T if all theparameters of T are ordered smaller than the leading variables of the polynomials in T . Theorem 8 ([32, Thm. 3.9]) . Let C = [ C , . . . , C r ] be the W-characteristic set of hF i ⊆ K [ x ] . Ifthe variable ordering condition is satisfied for C and C is not normal, then there exists an integerk (1 ≤ k < r ) such that [ C , . . . , C k ] is normal and [ C , . . . , C k + ] is not regular. Some structural properties about pseudo-divisibility among polynomials in the Gr¨obner basescan be found in [32]. Based on those properties, an e ff ective algorithm for normal triangulardecomposition of polynomial sets has been proposed in [33]. Definition 9.
A pair ( G , C ) of polynomial sets in K [ x ] is called a characteristic pair if G is aGr¨obner basis and C is the W-characteristic set of hGi . We say that the characteristic pair ( G , C )is strong if sat( C ) = hGi .A characteristic pair ( G , C ) is said to be regular or normal if C is regular or normal, respec-tively. Let ( { } , [1]) be regarded as a trivial regular / normal characteristic pair. If a characteristicpair is strong and regular, then the Gr¨obner basis G in the pair is said to be strong regular . Byregular or normal characteristic decomposition of a polynomial set F ⊆ K [ x ], we mean a finiteset of regular or normal characteristic pairs ( G , C ) , . . . , ( G e , C e ) satisfying the ideal relations in(3).The expression (3) can be rewritten in terms of the zero sets or varieties as Z ( F ) = e [ i = Z ( G i ) = e [ i = Z (sat( C i )) . (4)When the regular or normal characteristic pairs ( G i , C i ) are computed by the algorithms describedin [30, 33], the zero relation Z ( F ) = e [ i = Z ( C i / ini( C i )) (5)also holds.To compute a desired characteristic decomposition of a polynomial set F , we can firstproceed to decompose F into finitely many Gr¨obner bases G i of certain kinds and then formthe characteristic pairs ( G i , C i ) by simply extracting the W-characteristic sets C i from G i .6 . Computing strong regular characteristic pairs The main objective of this section is to show how to construct from an arbitrary ideal I ansrc pair ( G , C ) such that I ⊆ sat( C ) by computing Gr¨obner bases. The construction enables us todevise a novel algorithm for decomposing any polynomial set into finitely many src pairs. Definition 10.
A Gr¨obner basis G is said to be characterizable if hGi = sat( C ), where C is theW-characteristic set of hGi .Obviously, the W-characteristic set C extracted from the Gr¨obner basis G is unique. Thefollowing proposition shows that if sat( C ) = hGi , then C must be regular and thus ( G , C ) is ansrc pair. In other words, the Gr¨obner basis G in any src pair ( G , C ) is characterizable. Otherwise,sat( C ) , hGi ; in this case, G may not necessarily be determined by C (i.e., there may be twoGr¨obner bases G and G such that hG i and hG i have the same W-characteristic set C ). Proposition 11.
The W-characteristic set of the ideal generated by any characterizable Gr¨obnerbasis is regular.Proof.
Let G be any characterizable Gr¨obner basis and C = [ C , . . . , C r ] be the W-characteristicset of hGi . For any P ∈ sat( C ), we have P ∈ hGi since hGi = sat( C ). From Proposition 7(a) onecan see that prem( P , C ) =
0, which means that sat( C ) ⊆ { P ∈ K [ x ] : prem( P , C ) = } ⊆ sat( C ).Therefore, by [2, Thm. 6.1] C is regular. Theorem 12 (Strong Regularization) . Let C be the W-characteristic set of an arbitrary ideal I ⊂ K [ x ] , G i be the Gr¨obner basis of sat( C i − ) , and C i be the W-characteristic set of hG i i fori ≥ . Then there exists an integer m ≥ such that I ⊆ sat( C ) ⊆ sat( C m ) and either C m is aregular set, or sat( C m ) = h i .Proof. From Proposition 7(a) and sat( C i − ) = hG i i ⊆ sat( C i ) we know that sat( C ) ⊆ sat( C ) ⊆· · · ⊆ sat( C i ) ⊆ · · · for i ≥
2. Then by the Ascending Chain Condition there exists an m ≥ C m − ) = sat( C m ) = · · · . It follows that the Gr¨obner basis G m is characterizable unlesssat( C m ) = h i . Therefore, one sees from Proposition 11 that C m is regular or sat( C m ) = h i .The process of constructing the chain of W-characteristic sets C i of I i = hG i i for i = , . . . , m such that C m is regular and sat( C m ) = sat( C m − ) (called strong regularization ) shownin Theorem 12 is depicted by the diagram in Figure 1. I = I = hG i −→ C −→ sat( C ) ( , h i ) ∩ ∩ = I = hG i −→ C −→ sat( C ) ( , h i ) ∩ ∩ ... ... ∩ ∩ = I m − = hG m − i −→ C m − −→ sat( C m − ) ( , h i ) ∩ qq = I m = hG m i −→ C m −→ sat( C m ) = sat( C m + ) = · · · Figure 1: Strong regularization of W-characteristic sets by means of saturation emark 13. From [26, Thm. 4.4] we know that for any normal set T , if its parameters areordered smaller than the other variables, then the W-characteristic set C of sat( T ) is also normal.This corresponds in some way to the case of Theorem 12 when the variable ordering conditionis satisfied and C is assumed to be regular; in this case C is always regular and thus m = C is not necessarily regular (i.e., m may begreater than 2), as shown by Example 15 (where m = T without the variable ordering condition.For any triangular set T , sat( T ) is said to be equiprojectable if there exists a regular set ¯ T such that sat( T ) = sat( ¯ T ) [3]. The following corollary follows directly from [26, Thm. 4.4]. Corollary 14.
For any triangular set T satisfying the variable ordering condition, sat( T ) isequiprojectable if and only if the W-characteristic set C of sat( T ) is regular and sat( T ) = sat( C ) . Corollary 14 points out explicitly how to construct ¯ T from T and thus how to check whethersat( T ) is equiprojectable. Example 15.
Let C = [ x − x , ( y − x )( y − , ( y − z ] ⊆ K [ x , y , z ] with x < y < z . It is easyto verify that C is a Gr¨obner basis, it is a triangular set, and it is also the W-characteristic setof I = hC i , but C is not regular. The Gr¨obner basis of the saturated ideal of C may be easilycomputed as G = { z , x ( x − , y − x , x ( y + } . The W-characteristic set of hG i = sat( C ) is C = [ x ( x − , x ( y + , z ]. Obviously C is not regular and thus sat( C ) is not equiprojectable,while the W-characteristic set C = [ x − , y + , z ] of sat( C ) is regular. Therefore, for thisexample the integer m in Theorem 12 is equal to 3. Example 16.
Consider the regular set T = [ y , x z + xy ] in K [ x , y , z ] with y < x < z (cf. [33,Ex. 3.20]). The reduced lex Gr¨obner basis G of the saturated ideal sat( T ) of T is { y , yz , xz + y , z } ,so the W-characteristic set C of hGi is [ y , yz ]. One can easily see that C is not regular.Algorithm 1 can be used to compute an src pair from any polynomial ideal. Its terminationand correctness follow directly from Theorem 12. Algorithm 1: ( G , C ) : = srcPair ( I ) (for computing an src pair from a polynomial ideal) Input: I , an ideal in K [ x ]. Output: ( G , C ), an src pair such that I ⊆ sat( C ) = hGi , or ( { } , [1]) when I = h i . G : = Gr¨obner basis of I ; if G , { } then C : = W-characteristic set of hGi ; while sat( C ) , hGi do G : = Gr¨obner basis of sat( C ); if G = { } then return ( { } , [1]); C : = W-characteristic set of hGi ; else C : = [1]; return ( G , C ) 8 . Strong regular characteristic decomposition In this section, we present an algorithm to compute strong regular characteristic decomposi-tions of polynomial sets using Gr¨obner basis and ideal computations. The main ingredients ofthe algorithm are two subalgorithms: one for computing src pairs with ideal saturation and theother for computing src divisors with ideal quotient.For any two ideals I and J , we say that J divides I if I : J , I . Definition 17.
Let I be an ideal in K [ x ] and C be the W-characteristic set of I . C is said to be strong if sat( C ) = I , or morbid if sat( C ) does not divide I .Let C be the W-characteristic set of an ideal I ⊆ K [ x ]. If C is strong, then C carries allthe information of I . If C is morbid, then the associated primes of I are all properly containedin those of sat( C ); in this case the structure of I is so complicated that C carries almost noinformation of I . Proposition 18.
Let I and J be two ideals in K [ x ] such that I ⊆ J . Then √ I = √ J ∩ √ I : J .Proof. On one hand, for any polynomial P ∈ I , we have P ∈ J and P ∈ I : J since I ⊆ J and I ⊆ I : J . It follows that P ∈ J ∩ ( I : J ), which implies that √ I ⊆ √ J ∩ √ I : J . On the otherhand, for any polynomial P ∈ √ J ∩ √ I : J , we have P ∈ √ J and P ∈ √ I : J . As P ∈ √ J , thereexists an m such that P m ∈ J ; as P also belongs to √ I : J , one can find an l such that P l H ∈ I forany polynomial H ∈ J . Therefore, P l P m ∈ I and thus P ∈ √ I . Hence √ J ∩ √ I : J ⊆ √ I .For two ideals I and J in K [ x ] with I ⊆ J , by Proposition 18, √ I = √ J ∩ √ I : J . Thisdecomposition is trivial if I : J = I . For any given ideal I ( K [ x ], we will show in thefollowing subsection how to construct an ideal J ⊆ K [ x ] such that I ⊆ J and I : J , I . Withthe ideal J specially constructed from I , the splitting of I to J and I : J is nontrivial and √ J and √ I : J can be further decomposed. The technique of splitting ideals by taking quotientaccording to Proposition 18 originates from the idea of dividing a known subvariety out of anygiven variety explained in [31, pp. 196–197]. Definition 19.
An src pair ( G , C ) of polynomial sets is called an src divisor of a polynomial ideal I in K [ x ] if hGi = sat( C ) divides I .Given an ideal I ( K [ x ], we want to find an src divisor ( G , C ) of I . Suppose that I , h i and compute an src pair ( ¯ G , ¯ C ) : = srcPair ( I ) (using Algorithm 1) such that I ⊆ sat( ¯ C ). If I : sat( ¯ C ) , I , then ( G , C ) = ( ¯ G , ¯ C ) , ( { } , [1]) is an src divisor of I .Otherwise, I : sat( ¯ C ) = I . Let G ∗ be the Gr¨obner basis of I and extract the W-characteristicset C ∗ of I from G ∗ . Recall that I ⊆ sat( C ∗ ) and I , h i . If sat( C ∗ ) = I , then sat( ¯ C ) = sat( C ∗ )according to Theorem 12 (or Fig. 1), so I : sat( ¯ C ) = sat( C ∗ ) : sat( C ∗ ) = h i , I , whichcontradicts the assumption that I : sat( ¯ C ) = I . Therefore, sat( C ∗ ) , I and one can find apolynomial F ∈ sat( C ∗ ), F < I . Let J = Q C ∈C ∗ ini( C ). Then there exists a positive integer q such that F J q ∈ hC ∗ i ⊆ I . Now choose polynomials H , . . . , H t ∈ K [ x ] \ I such that H · · · H t = ⇐⇒ F J q = Natural choices for { H , . . . , H t } in line 7 of Algorithm 2 are { F , ini( C ) : C ∈ C ∗ , ini( C ) < K } and { H ∈ K [ x ] : H | F Q C ∈C ∗ ini( C ) , H is irreducible (or squarefree) } . Our experiments are done with the latter choice for squarefree H .
91) Let ( ˜ G , ˜ C ) , ( { } , [1]) be an src pair computed by Algorithm srcPair from h I ∪ { H i }i forsome 1 ≤ i ≤ t . If I : sat( ˜ C ) , I , then ( G , C ) = ( ˜ G , ˜ C ) is an src divisor of I .(2) Otherwise, sat( ˜ C ) does not divide I , where ( ˜ G , ˜ C ) is computed from h I ∪ { H i }i as in (1),for all 1 ≤ i ≤ t . In this case, proceed further to find an src divisor of I from h I ∪ { H i , H i j }i similarly, where H i j ∈ K [ x ] \h I ∪{ H i }i such that H i · · · H it i = ⇐⇒ F i Q C ∈C ∗ i ini( C ) = F i ∈ sat( C ∗ i ) \ h I ∪ { H i }i , and C ∗ i is the W-characteristic set of h I ∪ { H i }i .The above process can continue recursively until an src divisor of I is found. We formulatethe process as Algorithm 2. Algorithm 2: ( G , C ) : = srcDivisor ( I ) (for finding an src divisor of a polynomial ideal) Input: I , h i , an ideal in K [ x ]. Output: ( G , C ), an src divisor of I . ( G , C ) : = srcPair ( I ); ¯ I : = I : sat( C ); if ¯ I = I then G ∗ : = Gr¨obner basis of I ; C ∗ : = W-characteristic set of hG ∗ i ; Choose F ∈ sat( C ∗ ) \ I and H , . . . , H t ∈ K [ x ] \ I such that H · · · H t = ⇐⇒ F Q C ∈C ∗ ini( C ) = for i = , . . . , t do ( G , C ) : = srcPair ( h I ∪ { H i }i ); ¯ I : = I : sat( C ); if ¯ I , I then return ( G , C ) ; for i = , . . . , t do ( G , C ) : = srcDivisor ( h I ∪ { H i }i ); ¯ I : = I : sat( C ); if ¯ I , I then return ( G , C ); return ( G , C ) Proof of Termination and Correctness of Algorithm 2.
The process of searching for an src divi-sor of the ideal I in Algorithm 2 can be viewed as of building up a multi-branch tree Γ with I associated to its root, I j associated to its nodes of the first level, I j j associated to its nodes ofthe second level, etc. In general, the nodes j · · · j l of the l th level of Γ are associated with I j ··· j l .For each index j · · · j l there is an src pair ( G j ··· j l , C j ··· j l ) such that either (1) I j ... j l = h i , or (2) I : sat( C j ··· j l ) , I , or (3) I : sat( C j ··· j l ) = I . In case (1) or (2), the tree Γ terminates to growand the node j · · · j l becomes a leaf. When (2) happens, an src divisor of I is found. In case(3), new nodes j · · · j l i of the ( l + H j ··· j l i < I j ··· j l of F [ Q C ∈C ∗ j ··· jl ini( C )] q ∈ hC ∗ j ··· j l i ⊆ I j ··· j l , where F ∈ sat( C ∗ j ··· j l ) \ I j ··· j l , q is a positive integer, and C ∗ j ··· j l is the W-characteristic set of I j ··· j l . The new nodes are associated with h I j ··· j l ∪ { H j ··· j l i }i ( s j ··· j l ≤ i ≤ t j ··· j l ). 10Termination) As H j ··· j l i < I j ··· j l and I j ··· j l i = h I j ··· j l ∪ { H j ··· j l i }i , the containment I ⊂ I j ⊂· · · ⊂ I j ··· j l ⊂ I j ··· j l i is strict according to the Ascending Chain Condition for polynomial ideals.Hence the length of every branch of the tree Γ from the root to a leaf is finite, so the algorithmmust terminate.(Correctness) Recall Proposition 18 and observe that Q t j ··· jl i = s j ··· jl H j ··· j l i ∈ p I j ··· j l . Let ∆ be theset of indices of all the leaves of Γ . Then, after the growth of the tree Γ from a node is completed, √ I = \ δ ∈ ∆ p I δ always holds. Therefore, at the termination of the algorithm, either an src divisor ( G , C ) is found,or h i is associated to all the leaves of Γ ; the latter contradicts I , h i . The src divisor producedin line 12, 17, or 18 is what we want to find.The following theorem follows directly from Algorithm 2. Theorem 20.
From any ideal I , h i in K [ x ] , one can construct an src divisor of I .4.2. An ideal-division-based algorithm for src decomposition In this subsection, we describe an algorithm to decompose any polynomial set, or the ideal I it generates, into finitely many src pairs. Based on the strategy of ideal splitting with quotient,the algorithm works by finding an src divisor ( G , C ) of I and then dividing the saturated ideal of C out of I iteratively.Consider an ideal I , initially with input F ⊆ K [ x ] as its generating set of polynomials.Let Ψ be the set of src pairs already computed; it is ∅ initially. Suppose that I , h i andcompute ( G , C ) : = srcDivisor ( I ); in this case, I is split into sat( C ) and I : sat( C ) and an src pair( G , C ) , ( { } , [1]) is obtained and adjoined to Ψ . From Proposition 18 we know that √ I = p sat( C ) T √ I : sat( C ) , (6)so the procedure can continue to decompose I : sat( C ) instead of I .The above process of decomposition will terminate in finitely many iterations. We formulatethe process as Algorithm 3. Algorithm 3: Ψ : = srcDec ( F ) (for computing an src decomposition of a polynomial set) Input: F , a finite, nonempty set of polynomials in K [ x ]. Output: Ψ , an src decomposition of F such that √hF i = T C∈ Ψ √ sat( C ) = T G∈ Ψ √hGi ,or the empty set when hF i = h i . Ψ : = ∅ ; I : = hF i ; while I , h i do ( G , C ) : = srcDivisor ( I ); Ψ : = Ψ ∪ { ( G , C ) } ; I : = I : hGi ; return Ψ roof of Termination and Correctness of Algorithm 3. (Termination) Decomposition in Algori-thm 3 is an iterative process: every time after the ideal I is decomposed, a new ideal ¯ I ) I isobtained and then further decomposed in the same way. Since every new ideal generated in line6 is strictly enlarged, by the Ascending Chain Condition the algorithm terminates.(Correctness) When hF i = h i in line 3, Ψ = ∅ is returned. Hence we only need to showthat when hF i , h i , Ψ is an src decomposition of F , namely all the pairs in Ψ are src pairs andthe relation (3) holds. It is clear that only in line 5 is adjoined to Ψ a new pair ( G , C ) which iscomputed by Algorithm 2. Therefore, all the pairs in Ψ are src pairs.Now we prove that √hF i = T C∈ Ψ √ sat( C ). For those src pairs generated in line 5, one caneasily see that I ⊆ sat( C ); thus from Proposition 18 we know that the relation (6) holds.The relation (6) implies that every polynomial F ∈ √ I is in the intersection of √ sat( C ) and √ ¯ I , where ¯ I is a new ideal obtained which remains for further processing. This proves thatthe relation √hF i = T C∈ Ψ √ sat( C ) holds when the algorithm terminates with ¯ I = h i . Since hGi = sat( C ) for every src pair ( G , C ) ∈ Ψ , √hF i = T G∈ Ψ √hGi . This completes the proof of therelation (3).
5. Examples and experiments
Example 21.
Let F = { uxy , vy + y , vx + y } ⊆ K [ u , v , x , y ] with u < v < x < y . The Gr¨obnerbasis of I = hF i can be easily computed as G = { uvx , v x + vx , y − v x } and the W-characteristic set of hG i is C = [ uvx , y − v x ]. The saturated ideal of C is sat( C ) = h x , y i . One sees that ( ¯ G , ¯ C ) is an src pair, where ¯ G = { x , y } is the Gr¨obner basis and ¯ C = [ x , y ] is the W-characteristic set of sat( C ). It is easy to verify that I = I : sat( ¯ C ) = hG i is strictly larger than I , where G = { uv , vP , Q } is the Gr¨obner basis of I and P = v x + Q = y − v x . Hence ( ¯ G , ¯ C ) is adjoined to Ψ and the procedure continues to decompose I instead of I . The W-characteristic set C of I consists of the same polynomials as G : C = [ C , C , C ] = [ uv , vP , Q ] . Since sat( C ) = h i , the pair ( { } , [1]) is obtained and I : h i = I . As 1 ∈ sat( C ),ini( C ) · ini( C ) = v u = uC − v xC ∈ hC i , so we can take, e.g., H = v and H = u . Compute the Gr¨obner basis G = { v , y } of hG ∪ { H }i and extract the W-characteristic set C = [ v , y ] of hG i from G ; one finds that C is regular andsat( C ) = h v , y i = hG i and I : sat( C ) = h u , P , Q i is strictly larger than I . Then the src pair( G , C ) is obtained and adjoined to Ψ . The procedure continues with I = h u , P , Q i .Simple computation shows that the Gr¨obner basis and the W-characteristic set of I containthe same polynomials: G = { u , P , Q } , C = [ u , P , Q ] . G , C ) is an src pair and I : sat( C ) = h i . Therefore, ( G , C ) is added to Ψ and the procedure terminates. Finally, an src decomposition Ψ = { ( ¯ G , ¯ C ) , ( G , C ) , ( G , C ) } of F is obtained. Example 22.
Consider the polynomial set F = {− ct u + t − uv − uw , − ct v + t − u v − vw , − ct w + t − u w − v w } (which is Ex 9 in Table 1) with variable ordering w < v < u < t < c . The ideal generated by F , which is of dimension 2 and not radical, consists of 8 primary components (none of them isembedded). The polynomial set F can be decomposed by Algorithm 3 into 6 src pairs( G , C ) = ( { v , u , t } , [ v , u , t ]) , ( G , C ) = ( { w , u , t } , [ w , u , t ]) , ( G , C ) = ( { w , v , t } , [ w , v , t ]) , ( G , C ) = ( { v − w , u − w , wt c − t + w } , [ v − w , u − w , wt c − t + w ]) , ( G , C ) = ( { v − w , P , Q , t c − wu + w } , [ v − w , P , Q ]) , ( G , C ) = ( { G , . . . , G } , [ G , G , G ]) , where the polynomials G , . . . , G consist of 6, 3, 4, 10, 3 terms respectively and P = t − wu − w u , Q = u c + wuc − ut − wt . It may be observed that (1) C i is normal for all i , (2) √hG i i is prime for i = , . . . ,
5, and (3) √hG i is composed of 3 prime ideals.The cyclic- n systems are well-known examples for which triangular decompositions basedon pseudo-division are more di ffi cult to compute than Gr¨obner bases. Our algorithm can computesrc pairs for cyclic-6 in a few minutes, while other triangular decomposition algorithms cannot(see Ex 21 in Table 1). We have implemented Algorithm 3 as a M aple function srcDec and carried out experimentswith the implementation on an Intel(R) Core(TM) i5-4210U CPU at 1.70 GHz × FGb library and M aple ’s built-in packages. Selected resultsof experiments on some test examples are presented in Table 1: Ex 1–4 are taken from theEpsilon library, Ex 5–9 from [27], Ex 10 from [8], Ex 11 from the
FGb library, and Ex 12–16can be found at , Ex 17–21at http://homepages.math.uic.edu/~jan/Demo/TITLES.html , and Ex 22–27 at . The function srcDec is implemented fordirect decomposition of polynomial sets into src pairs. To observe the performance of ouralgorithm, we made comparative experiments on srcDec in M aple
18 with two other relevantfunctions for unmixed decomposition of polynomial sets into Gr¨obner bases of saturated idealsof triangular sets: one is the
Epsilon function uvd which is implemented for decomposing anarbitrary algebraic variety into unmixed subvarieties. The other function computes first thedecomposition of a polynomial set into regular sets using the
RegularChains function
Trian-gularize and then the Gr¨obner bases of the saturated ideals of the computed regular sets.13n Table 1, Label indicates the label used in the above-cited references and Var, Pol, and Dimdenote the number of variables, the number of polynomials in the example, and the dimensionof the ideal generated by the polynomials, respectively. Total, GB, SAT, and QUO under Algo-rithm 3 record respectively the total time (followed by the number of src pairs in parentheses) forsrc decomposition using Algorithm 3, the time for computing all the Gr¨obner bases, saturatedideals, and ideal quotients; Total under uvd records the total time for unmixed decomposition.Total and Regular under
Triangularize record the total time for unmixed decomposition (fol-lowed by the number of components in parentheses) and the time for regular decompositionrespectively.
Table 1: Timings for src decomposition (in second)Algorithm 3 uvd Triangularize
Ex Label Var Pol Dim Total GB SAT QUO Total Total Regular1 E1 10 10 1 1.450(2) 0.564 0.221 0.632 0.867(3) 2.041(13) 1.7092 E5 15 17 4 3.568(1) 1.793 0.461 1.255 3.374(1) 7.684(7) 5.2253 E11 4 3 1 0.037(1) 0.008 0.008 0.015 0.122(1) 0.079(3) 0.0354 E34 14 16 0 0.574(0) 0.556 0. 0. > > > > > > > > > > > > > The time-consuming steps in algorithm srcDec are for the computation of the Gr¨obner basesof the input ideals, the Gr¨obner bases of ideal quotients, and the Gr¨obner bases of saturatedideals. The regular sets computed by the algorithm are normal in most cases and they are oftensimpler than the corresponding regular sets computed by pseudo-division or subresultant-basedalgorithms, so that the Gr¨obner bases of the saturated ideals of the regular sets produced byour algorithm srcDec tend to be easier to compute than those produced by uvd or Triangular-ize for about two thirds of the test examples. More importantly, the number of src pairs inan src decomposition computed by srcDec is usually smaller than the number of componentsin the corresponding unmixed decomposition computed by uvd or regular decomposition by
Triangularize , as shown by the experimental data in Table 1. This is because our ideal-division-based algorithm generates few redundant components. For example, the computation of thesrc decomposition using srcDec takes much less time than that of the regular decomposition14sing
Triangularize for Ex 14, as the src decomposition contains only two src pairs.
6. Conclusion
In this paper, it is shown that a strong regular characteristic (src) pair, and therefore ansrc divisor ( G , C ) of I , can be constructed from any polynomial ideal I with given generatingset F . The constructed src divisor may be used to split the ideal I into the saturated ideal sat( C )and the quotient ideal I : sat( C ). The process of construction and splitting can be repeatedfor I : sat( C ) instead of I and recursively, yielding an algorithm capable of decomposing thepolynomial set F into finitely many src pairs ( G , C ) , . . . , ( G e , C e ) such that Z ( F ) = e [ i = Z ( G i ) = e [ i = Z (sat( C i )) (7)or equivalently (3) holds. The relation (7) provides two representations for the zero set of F : onein terms of the Gr¨obner bases G , . . . , G e and the other in terms of the regular sets C , . . . , C e .Several nice properties about strong regular characteristic pairs and characteristic decompositionshave been presented, and the implementation and performance of our proposed algorithm havebeen illustrated by examples and experimental results.The contributions of this paper include: (1) two main theorems (Theorems 12 and 20)showing how to construct an src pair and an src divisor of an arbitrary polynomial ideal; (2)an algorithm for decomposing any polynomial set F into src pairs ( G i , C i ) such that (3) and (7)hold; (3) some experiments with a preliminary implementation of the decomposition algorithm.The triangular sets in an src decomposition are normal in most cases (cf. [33, 32]). Itturns out that comprehensive triangular decompositions [8] and / or Gr¨obner systems [24] can bereproduced rather easily from src pairs computed by our algorithm, and we are working on thedetails. The W-characteristic set of an ideal may be morbid and it is not yet clear when morbidityhappens. How to establish equivalent conditions for a W-characteristic set to be morbid and howto retrieve information of an ideal from its morbid W-characteristic set are some of the questionsthat remain for further investigation.The authors wish to thank the referees for their insightful comments which helped bring thepaper to the present form. This work also benefited from frequent discussions the authors hadwith Chenqi Mou. References [1] P. Alvandi, C. Chen, S. Marcus, M. Moreno Maza, ´E. Schost, and P. Vrbik. Doing algebraic geometry with theRegularChains library. In H. Hong and C. K. Yap, editors,
Proceedings of ICMS 2014 , pages 472–479. Springer-Verlag, Berlin Heidelberg, 2014.[2] P. Aubry, D. Lazard, and M. Moreno Maza. On the theories of triangular sets.
Journal of Symbolic Computation ,28(1–2):105–124, 1999.[3] P. Aubry and A. Valibouze. Using Galois ideals for computing relative resolvents.
Journal of SymbolicComputation , 30(6):635–651, 2000.[4] T. Bchler, V. Gerdt, M. Lange-Hegermann, and D. Robertz. Algorithmic Thomas decomposition of algebraic anddi ff erential systems. Journal of Symbolic Computation , 47(10):1233–1266, 2012.[5] T. Becker, V. Weispfenning, and H. Kredel.
Gr¨obner Bases: A Computational Approach to Commutative Algebra .Graduate Texts in Mathematics. Springer, New York, 1993.[6] B. Buchberger.
Ein Algorithmus zum Au ffi nden der Basiselemente des Restklassenrings nach einem nulldimension-alen Polynomideal . PhD thesis, Universit¨at Innsbruck, Austria, 1965.
7] B. Buchberger. Gr¨obner bases: An algorithmic method in polynomial ideal theory. In N. Bose, editor,
Multidimensional Systems Theory , pages 184–232. Springer, Dordrecht, 1985.[8] C. Chen, O. Golubitsky, F. Lemaire, M. Moreno Maza, and W. Pan. Comprehensive triangular decomposition. InV. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, editors,
Proceedings of CASC 2007 , pages 73–101. Springer-Verlag, Berlin Heidelberg, 2007.[9] C. Chen and M. Moreno Maza. Algorithms for computing triangular decompositions of polynomial systems.
Journal of Symbolic Computation , 47(6):610–642, 2012.[10] D. Cox, J. Little, and D. O’Shea.
Ideals, Varieties, and Algorithms: An Introduction to Computational AlgebraicGeometry and Commutative Algebra . Undergraduate Texts in Mathematics. Springer, New York, 1997.[11] D. Cox, J. Little, and D. O’Shea.
Using Algebraic Geometry . Graduate Texts in Mathematics. Springer, New York,1998.[12] X. Dahan. On lexicographic Gr¨obner bases of radical ideals in dimension zero: Interpolation and structure. Preprintat arXiv:1207.3887, 2012.[13] J.-C. Faug`ere, P. Gianni, D. Lazard, and T. Mora. E ffi cient computation of zero-dimensional Gr¨obner bases bychange of ordering. Journal of Symbolic Computation , 16(4):329–344, 1993.[14] S. Gao, F. Volny, and M. Wang. A new framework for computing Gr¨obner bases.
Mathematics of Computation ,85(297):449–465, 2016.[15] X.-S. Gao and S.-C. Chou. On the dimension of an arbitrary ascending chain.
Chinese Science Bulletin , 38(10):799–804, 1993.[16] M. Kalkbrener.
Three Contributions to Elimination Theory . PhD thesis, Johannes Kepler University, Austria, 1991.[17] M. Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties.
Journal of Symbolic Computation , 15(2):143–167, 1993.[18] D. Lazard. Ideal bases and primary decomposition: Case of two variables.
Journal of Symbolic Computation ,1(3):261–270, 1985.[19] D. Lazard. A new method for solving algebraic systems of positive dimension.
Discrete Applied Mathematics ,33(1–3):147–160, 1991.[20] D. Lazard. Solving zero-dimensional algebraic systems.
Journal of Symbolic Computation , 13(2):117–131, 1992.[21] B. Li and D.-K. Wang. An algorithm for transforming regular chain into normal chain. In D. Kapur, editor,
Proceedings of ASCM 2007 , pages 236–245. Springer-Verlag, Berlin Heidelberg, 2008.[22] X. Li, C. Mou, and D. Wang. Decomposing polynomial sets into simple sets over finite fields: The zero-dimensionalcase.
Computers and Mathematics with Applications , 60(11):2983–2997, 2010.[23] M. G. Marinari and T. Mora. A remark on a remark by Macaulay or enhancing Lazard structural theorem.
Bulletinof the Iranian Mathematical Society , 29(1):1–45, 2003.[24] A. Montes. A new algorithm for discussing Gr¨obner bases with parameters.
Journal of Symbolic Computation ,33(2):183–208, 2002.[25] M. Moreno Maza. On triangular decompositions of algebraic varieties. Technical Report 4 /
99, NAG, UK, 2000.Presented at the MEGA-2000 Conference, Bath, UK.[26] C. Mou and D. Wang. Characteristic decomposition: From regular sets to normal sets.
Journal of Systems Scienceand Complexity , 32(1):37–46, 2019.[27] C. Mou, D. Wang, and X. Li. Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case.
Theoretical Computer Science , 468:102–113, 2013.[28] J. F. Ritt. Di ff erential Algebra . American Mathematical Society, New York, 1950.[29] T. Shimoyama and K. Yokoyama. Localization and primary decomposition of polynomial ideals. Journal ofSymbolic Computation , 22(3):247–277, 1996.[30] D. Wang. Computing triangular systems and regular systems.
Journal of Symbolic Computation , 30(2):221–236,2000.[31] D. Wang.
Elimination Methods . Springer-Verlag, Wien, 2001.[32] D. Wang. On the connection between Ritt characteristic sets and Buchberger-Gr¨obner bases.
Mathematics inComputer Science , 10:479–492, 2016.[33] D. Wang, R. Dong, and C. Mou. Decomposition of polynomial sets into characteristic pairs.
Mathematics ofComputation , 89(324):1993–2015, 2020.[34] W.-T. Wu. Basic principles of mechanical theorem proving in elementary geometries.
Journal of AutomatedReasoning , 2(3):221–252, 1986. ppendix A. A.1. Strong regular characteristic decomposition of radical ideals
Lemma 23.
Let I and J be two radical ideals in K [ x ] . Then the algebraic varieties Z ( I : J ) and Z ( J ) do not have any common irredundant irreducible component.Proof. Let Z ( I : J ) = V ∪ · · · ∪ V s be an irredundant irreducible decomposition of the algebraicvariety Z ( I : J ). Suppose that Z ( I : J ) and Z ( J ) have a common irredundant irreduciblecomponent, say V m for some positive integer m ( ≤ s ), and let V ∗ = [ ≤ i ≤ si , m V i . Then Z ( I ) \ Z ( J ) ⊆ V ∗ ( V ∪ · · · ∪ V s = Z ( I : J ); in this case, Z ( I ) \ Z ( J ) ⊂ V ∗ implies that Z ( I ) \ Z ( J ) ⊆ V ∗ = V ∗ , which contradicts with the fact that Z ( I : J ) = Z ( I ) \ Z ( J ). Proposition 24.
Let I and J be two radical ideals in K [ x ] . Then: ( a ) the saturated ideal of the W-characteristic of I is radical; ( b ) the ideal quotient I : J is also radical.Proof. ( a ) Let T be the W-characteristic set of I . To show that sat( T ) is radical, it su ff ers toshow that sat( T ) = √ sat( T ). The inclusion sat( T ) ⊆ √ sat( T ) is obvious. For any P ∈ √ sat( T ),we have P l I m ∈ hT i ⊆ I for some nonnegative integers m and l , where I = Q T ∈T ini( T ). Thus( IP ) max( m , l ) ∈ I ; it follows that IP ∈ √ I = I , so that prem( IP , T ) =
0, which means that P ∈ sat( T ). Therefore, sat( T ) = √ sat( T ).( b ) To prove that I : J is radical, we only need to prove that √ I : J ⊆ I : J . For any F ∈ √ I : J and J ∈ J , there exists a nonnegative integer l such that F l J ∈ I ; it follows that( F J ) l ∈ I , so that F J ∈ √ I = I . Hence F ∈ I : J .By Proposition 24, for any radical ideal I the saturated ideal sat( C ) of every regular set C appearing in an src decomposition Ψ of I is radical. According to Lemma 23, for any twosrc pairs ( G i , C i ) and ( G j , C j ) ( i , j ) in Ψ the two varieties Z (sat( C i )) and Z (sat( C j )) do not haveany common irredundant irreducible component. In other words, the ideals sat( C i ) and sat( C j )have no common minimal associated prime. Therefore, the number of src pairs in Ψ must besmaller than that of irredundant irreducible components of the variety Z ( F ). We are unable toestablish such a tight bound for nonradical ideals because when I and J in Lemma 23 are notradical, I : J and J may have a common minimal associated prime. Taking I = h x y i and J = h x i as an example, we have I : J = h x y i , which has a minimal associated prime h x i as J . A.2. Triangular decomposition and characteristic decomposition