Decomposition of polynomial sets into characteristic pairs
aa r X i v : . [ c s . S C ] F e b Decomposition of polynomial sets into characteristic pairs ∗ Dongming Wang ab , Rina Dong a , and Chenqi Mou † aa LMIB – SKLSDE – School of Mathematics and Systems Science,Beihang University, Beijing 100191, China { rina.dong, chenqi.mou } @buaa.edu.cn b Centre National de la Recherche Scientifique,75794 Paris cedex 16, [email protected]
Abstract
A characteristic pair is a pair ( G , C ) of polynomial sets in which G is a reducedlexicographic Gr¨obner basis, C is the minimal triangular set contained in G , and C isnormal. In this paper, we show that any finite polynomial set P can be decomposedalgorithmically into finitely many characteristic pairs with associated zero relations,which provide representations for the zero set of P in terms of those of Gr¨obnerbases and those of triangular sets. The algorithm we propose for the decompositionmakes use of the inherent connection between Ritt characteristic sets and lexicographicGr¨obner bases and is based essentially on the structural properties and the computa-tion of lexicographic Gr¨obner bases. Several nice properties about the decompositionand the resulting characteristic pairs, in particular relationships between the Gr¨obnerbasis and the triangular set in each pair, are established. Examples are given toillustrate the algorithm and some of the properties. Key words:
Characteristic pair, normal triangular set, lexicographic Gr¨obner basis, zero decom-position
Mathematics Subject Classification:
Systems of polynomial equations are fundamental objects of study in mathematics which occurin many domains of science and engineering. Such systems may be triangularized by usingthe well-known method of Gaussian elimination when the equations are linear. There are twoapproaches, developed on the basis of characteristic sets [35, 47] and Gr¨obner bases [7, 12], whichcan be considered as generalizations of Gaussian elimination to the case where the equations arenonlinear. Following these approaches of triangularization, the present paper is concerned withthe problem of decomposing an arbitrary set P of multivariate polynomials into finitely manytriangular sets of polynomials that may be used to represent the set of zeros of P (or equivalentlythe algebraic variety defined by P , or the radical of the ideal generated by P ). This problem of ∗ This work was supported partially by the National Natural Science Foundation of China (NSFC11401018) and the project SKLSDE-2015ZX-18. † Corresponding author: +8613811426823, [email protected], School of Mathematics andSystems Science, Beihang University, Beijing 100191, China. riangular decomposition is conceptually simple, but computationally difficult, and to it satisfactoryalgorithmic solutions are of both theoretical interest and practical value. The last three decadeshave witnessed extensive research on polynomial elimination and triangular decomposition, whichled to significant developments on the theories, methods, and software tools for polynomial systemsolving (see, e.g., [3, 4, 5, 8, 9, 10, 20, 23, 25, 28, 34, 37, 38, 39, 46, 48] and references therein).Along with these developments, triangular decomposition has become a standard approach tostudying computational problems in commutative algebra and algebraic geometry, a basic toolkitfor building advanced functions in modern computer algebra systems, and a general and powerfultechnique of breaking complex polynomial systems down into simply structured, easily manageablesubsystems for diverse scientific and engineering applications.To make our statements precise, we fix an order for the variables of the polynomials in question.A triangular set T is meant an ordered set of polynomials whose greatest variables strictly increasewith respect to the fixed variable order. T is said to be normal or called a normal set if noneof the greatest variables occurs in the leading coefficients of the polynomials in T with respect totheir greatest variables. By a polynomial system we mean a pair [ P , Q ] of polynomial sets withwhich the system of polynomial equations P = 0 and inequations Q 6 = 0 is of concern. It is calleda triangular system or a normal system , respectively, if P is a triangular set or a normal set and Q satisfies certain subsidiary conditions.Effective algorithms are now available for decomposing arbitrary polynomial sets or systemsof moderate size into triangular sets or systems of various kinds (regular, simple, irreducible, etc.)[40, 42, 25, 31], though it is not yet clear how to measure the quality of triangular decompositionsand how to produce triangular sets or systems of high quality in terms of theoretical properties(such as uniqueness, squarefreeness, and normality) and simplicity of expression (with lower degree,smaller size, and fewer components, etc.). One way to obtain “good” triangular decompositionsis via computation of lexicographic (lex) Gr¨obner bases, where the lex term ordering determinedby the variable order ensures that the bases have certain triangular structures with nice algebraicproperties [6, 12]. For the zero-dimensional case, relationships between Gr¨obner bases and trian-gular sets were studied in [29], leading to algorithms for the computation of triangular sets fromlex Gr¨obner bases based on factorization and the D5 principle [14]. More recently, an algorithm fortriangular decomposition of zero-dimensional polynomial sets has been proposed in [13], based onan exploration of the structures of lex Gr¨obner bases. For polynomial ideals of arbitrary dimension,the connection between Ritt characteristic sets and lex Gr¨obner bases has been investigated in [43];(pseudo-) divisibility relationships established therein will be clarified and used in later sections.The structures of lex Gr¨obner bases were studied first by Lazard [27] for bivariate ideals andthen extended to general zero-dimensional (radical, multivariate) ideals in a number of papers[24, 21, 32, 13] with many deep results.One kind of presumably good triangular sets is normal sets explained above, which appeared forthe first time as normalized triangular sets in [28] and later as p-chains in [20], and were elaboratedin [40, Sect. 5.2] and [15]. Normal sets and systems enjoy a number of remarkable properties and areconvenient for various applications, in particular dealing with parametric polynomial systems [8,20, 40]. There are algorithms for normalizing triangular sets, and more generally, for decomposingarbitrary polynomial sets or systems into normal sets or systems [40, 44, 31].In this paper, we focus our study on what we call characteristic pair and characteristicdecomposition : the former is a pair ( G , C ) of polynomial sets in which G is a reduced lex Gr¨obnerbasis, C is the minimal triangular set contained in G , and C is normal; the latter is the decompositionof a finite polynomial set P into finitely many characteristic pairs with associated zero relations,which provide representations for the zero set of P in terms of those of Gr¨obner bases and thoseof triangular sets. Our main contributions include: (1) clarification of the connection betweennormal sets and lex Gr¨obner bases via the concept of W-characteristic sets (introduced in [43]), (2)introduction of the concepts of (strong) characteristic pairs and characteristic decomposition withseveral properties proved, (3) an algorithm for computing (strong) characteristic decompositionsof polynomial sets, and (4) experimental results illustrating the performance of our algorithm andits implementation.The proposed algorithm, which makes use of the inherent connection between characteristic setsand Gr¨obner bases for splitting, is capable of decomposing any given polynomial set simultaneouslyinto finitely many normal sets C , . . . , C t and lex Gr¨obner bases G , . . . , G t with every C i contained in i . It is proved that each C i can be reduced to a Ritt characteristic set of the ideal generated by G i if it is not reduced (Theorem 3.14 and Corollary 3.15). It is also shown that a strong characteristicdecomposition can be computed out of any characteristic decomposition without need of furthersplitting (Theorems 3.19 and 3.22).After a brief review of Gr¨obner bases, normal sets, and W-characteristic sets in Section 2, wewill define (strong) characteristic pairs and (strong) characteristic decomposition and prove someof their properties in Section 3, describe the decomposition algorithm with proofs of terminationand correctness in Section 4, and illustrate how the algorithm works with an example and reportour experimental results in Section 5. We recall some basic notions and notations which will be used in later sections and highlight theintrinsic structures of reduced lex Gr¨obner bases on which the main results of this paper are based.For more details about the theories of Gr¨obner bases (also called Buchberger-Gr¨obner bases) andtriangular sets, the reader is referred to [3, 12, 40] and references therein.
A Gr¨obner basis of a polynomial ideal is a special set of generators of the ideal which is wellstructured and has good properties. The structures and properties of Gr¨obner bases allow one tosolve various computational problems with polynomial ideals, such as basic ideal operation, idealmembership test, and primary ideal decomposition. Introduced by Buchberger [6] in 1965 andhaving been developed for over half a century [7, 22, 45, 18, 36, 16, 17, 19, 26], Gr¨obner bases havebecome a truly powerful method that has applications everywhere polynomial ideals are involved.Let K be any field and K [ x , . . . , x n ] be the ring of polynomials in the variables x , . . . , x n with coefficients in K . In the sequel, we fix the variable order as x < · · · < x n unless otherwisespecified. For the sake of simplicity, we write x for ( x , . . . , x n ), x i for ( x , . . . , x i ), and K [ x ] for K [ x , . . . , x n ].A total ordering < on all the terms in K [ x ] is called a term ordering if it is a well ordering andfor any terms u , v , and w in K [ x ], u > v implies uw > vw . In this paper we are concerned mainlywith the lex term ordering, with respect to which Gr¨obner bases possess rich algebraic structures.For any two terms u = x α and v = x β in K [ x ], we say that u > lex v if the left rightmost nonzeroentry in the vector α − β is positive.Fix a term ordering < . The greatest term in a polynomial F ∈ K [ x ] with respect to < iscalled the leading term of F and denoted by lt( F ). As usual, h{ F , . . . , F s , . . . }i = h F , . . . , F s , . . . i denotes the ideal generated by the polynomials F , . . . , F s , . . . ∈ K [ x ]. Definition 2.1
Let I ⊆ K [ x ] be an ideal and < be a term ordering. A finite set { G , . . . , G s } ofpolynomials in I is called a Gr¨obner basis of I with respect to < if h lt( G ) , . . . , lt( G s ) i = h lt( I ) i ,where lt( I ) denotes the set of leading terms of all the polynomials in I .Let G = { G , . . . , G s } be a Gr¨obner basis of an ideal I ⊆ K [ x ] with respect to a fixed termordering < . For any polynomial F ∈ K [ x ], there is a unique polynomial R ∈ K [ x ] corresponding to F such that F − R ∈ I and no term of R is divisible by any of lt( G ) , . . . , lt( G s ). The polynomial R is called the normal form of F with respect to G (denoted by nform( F, G )), and F is said tobe B-reduced with respect to G if F = R . The Gr¨obner basis G itself is said to be reduced if thecoefficient of each G i in lt( G i ) is 1 and no term of G i lies in h{ lt( G ) | G ∈ G , G = G i }i for all i = 1 , . . . , s . The reduced Gr¨obner basis of I with respect to a fixed term ordering is unique. Example 2.2
Consider P = { x x , x x + x x , x , x x , ( x x +1) x + x x x } ⊆ K [ x , x , x , x ].The polynomial set G = { x x , x x , x , x } is a Gr¨obner basis of the ideal hPi with respect to thelex ordering on x < x < x < x . One can easily check that G is also the reduced lex Gr¨obnerbasis of P . Remark 2.3
The term
B-reduced is an abbreviation of
Buchberger-reduced for a polynomialmodulo a Gr¨obner basis. We use the prefix B to distinguish this term from the term
R-reduced (short for Ritt-reduced, defined below) for a polynomial modulo a triangular set. .2 Normal triangular sets Now let F be a polynomial in K [ x ] \ K . With respect to the variable order, the greatest variablewhich actually appears in F is called the leading variable of F and denoted by lv( F ). Let lv( F ) = x i ; then F can be written as F = Ix ki + R , with I ∈ K [ x i − ], R ∈ K [ x i ], and deg( R, x i ) < k =deg( F, x i ). The polynomial I is called the initial of F , denoted by ini( F ). For any polynomial set F ⊆ K [ x ], ini( F ) denotes { ini( F ) | F ∈ F} . Definition 2.4
Any finite, nonempty, ordered set [ T , . . . , T r ] of polynomials in K [ x ] \ K is calleda triangular set if lv( T ) < · · · < lv( T r ) with respect to the variable order.The saturated ideal of a triangular set T = [ T , . . . , T r ] is defined as sat( T ) = hT i : J ∞ , where J = ini( T ) · · · ini( T r ). We write sat i ( T ) = sat([ T , . . . , T i ]) for i = 1 , . . . , r . The variables in { x , . . . , x n } \ { lv( T ) , . . . , lv( T r ) } are called the parameters of T . A triangular set T is said to be zero-dimensional if there is no parameter of T , and positive-dimensional otherwise. Definition 2.5
A triangular set T = [ T , . . . , T r ] ⊆ K [ x ] is said to be normal (or called a normalset ) if all ini( T ) , . . . , ini( T r ) involve only the parameters of T . Example 2.6
From the Gr¨obner basis G = { x x , x x , x , x } in Example 2.2 one can extract twotriangular sets T = [ x x , x x , x ] and T = [ x x , x , x ]. Both of them are positive-dimensional,with x as their parameter. One can easily see that T is not normal, but T is.Among the most commonly used triangular sets there are regular sets [39] or regular chains [3]. A triangular set T = [ T , . . . , T r ] ⊆ K [ x ] is called a regular set or said to be regular if forevery i = 2 , . . . , r , ini( T i ) is neither zero nor a zero-divisor in K [ x ] / sat i − ( T ). By definition anynormal set is obviously regular. It is proved in [3, 40] that a triangular set T is regular if and onlyif sat( T ) = { F | prem( F, T ) = 0 } .A nonzero polynomial P ∈ K [ x ] is said to be R-reduced with respect to Q ∈ K [ x ] \ K ifdeg( P, lv( Q )) < deg( Q, lv( Q )); P is R-reduced with respect to a triangular set T = [ T , . . . , T r ] ⊆ K [ x ] if P is R-reduced with respect to all T i for i = 1 , . . . , r . A triangular set T itself is said to be R-reduced if T i is R-reduced with respect to [ T , . . . , T i − ] for all i = 2 , . . . , r .Denote by prem( P, Q ) the pseudo-remainder and by pquo(
P, Q ) the pseudo-quotient of P ∈ K [ x ] with respect to Q ∈ K [ x ] \ K in lv( Q ), and for any triangular set T = [ T , . . . , T r ] ⊆ K [ x ]define prem( P, T ) = prem( · · · prem(prem( P, T r ) , T r − ) , . . . , T ) , called the pseudo-remainder of P with respect to T . Clearly, prem( P, Q ) and prem( P, T ) arerespectively R-reduced with respect to Q and T . Similarly, we can define the resultant of P withrespect to T as res( P, T ) = res( · · · res(res( P, T r ) , T r − ) , . . . , T ) , where res( P, Q ) denotes the resultant of P ∈ K [ x ] and Q ∈ K [ x ] \ K with respect to lv( Q ). From the reduced lex Gr¨obner basis G of any polynomial ideal hPi ⊆ K [ x ], one can extract theW-characteristic set C of hPi defined below. Definition 2.7 ([43, Def. 3.1])
Let G be the reduced lex Gr¨obner basis of an ideal generated byan arbitrary polynomial set P ⊆ K [ x ], and denote by G ( i ) = { G ∈ G| lv( G ) = x i } . Then the set n [ i =1 { G ∈ G ( i ) | ∀ G ′ ∈ G \ { G } , G < lex G ′ } , ordered according to < lex , is called the W-characteristic set of hPi .The set in Definition 2.7 is also called the W-characteristic set of the reduced lex Gr¨obner basis G for the sake of simplicity. By definition any W-characteristic set is a triangular set. xample 2.8 Clearly among the two triangular sets T and T extracted from the reduced lexGr¨obner basis G in Example 2.6, T is the W-characteristic set of hPi , for x x < lex x . Definition 2.9
Let P be any finite polynomial set in K [ x ]. An R-reduced triangular set C ⊆ K [ x ]is called a Ritt characteristic set of the ideal hPi if C ⊆ hPi and for any P ∈ hPi , prem( P, C ) = 0.The W-characteristic set C of hPi is a Ritt characteristic set of hPi if C is R-reduced [43,Thm. 3.3]. Further relationships between Ritt characteristic sets and lex Gr¨obner bases are estab-lished in [43] with the help of the concept of W-characteristic sets.For any polynomial P ∈ K [ x ] and polynomial set P ⊆ K [ x ], we denote by Z ( P ) the set ofzeros of P in ¯ K , the algebraic closure of K , and by Z ( P ) the set of common zeros of all thepolynomials in P in ¯ K n . For any nonempty polynomial sets P and Q in K [ x ], we define Z ( P / Q ) = Z ( P ) \ Z (Π Q ∈Q Q ). Proposition 2.10 ([43, Prop. 3.1])
Let C be the W-characteristic set of hPi ⊆ K [ x ] . Then ( a ) for any P ∈ hPi , prem( P, C ) = 0 ; ( b ) hCi ⊆ hPi ⊆ sat( C ) ; ( c ) Z ( C / ini( C )) ⊆ Z ( P ) ⊆ Z ( C ) . The following theorem [43, Thm. 3.9] exploits the pseudo-divisibility relationships betweenpolynomials in W-characteristic sets (and thus between those in reduced lex Gr¨obner bases) forpolynomial ideals of arbitrary dimension, while other well-known structural properties of lexGr¨obner bases were established only for bivariate or zero-dimensional polynomial ideals. It isthese relationships that enable us to adopt an effective splitting strategy for our algorithm ofcharacteristic decomposition.
Theorem 2.11 ([43, Thm. 3.9])
Let C = [ C , . . . , C r ] be the W-characteristic set of hPi ⊆ K [ x ] .If C is not normal, then there exists an integer k (1 ≤ k ≤ r ) such that [ C , . . . , C k ] is normal and [ C , . . . , C k +1 ] is not regular.Assume that the variables x , . . . , x n are ordered such that the parameters of C are all smallerthan the other variables and let I k +1 = ini( C k +1 ) and l be the integer such that lv( I k +1 ) = lv( C l ) . ( a ) If I k +1 is not R-reduced with respect to C l , then prem( I k +1 , [ C , . . . , C l ]) = 0 , prem( C k +1 , [ C , . . . , C k ]) = 0 . ( b ) If I k +1 is R-reduced with respect to C l , then prem( C l , [ C , . . . , C l − , I k +1 ]) = 0 and either res(ini( I k +1 ) , [ C , . . . , C l − ]) = 0 or prem( C k +1 , [ C , . . . , C l − , I k +1 , C l +1 , . . . , C k ]) = 0 . Example 2.12
The W-characteristic set T = [ x x , x x , x ] of hPi in Example 2.8 is not normal,and one can find that [ x x ] is normal, but [ x x , x x ] is not (furthermore, it is not regular). Theinitial x of x x is not R-reduced with respect to x x , and one can check that prem( x , [ x x ]) = 0and prem( x x , [ x x ]) = 0, which accord with Theorem 2.11(a).An obvious consequence of the first part of the theorem is that the W-characteristic setcontained in the reduced lex Gr¨obner basis of a polynomial ideal, if it is regular, must be normal.This implies that certain normalization mechanism is integrated into the algorithm of Gr¨obnerbases, so that triangular subsets of lex Gr¨obner bases are normalized as much as possible.The condition on the order of x , . . . , x n for C in Theorem 2.11 is needed, for otherwise thetheorem does not necessarily hold, as shown by [43, Ex. 3.1(b)]. In the latter case, one may changethe variable order properly to make the condition satisfied, so as to obtain the pseudo-divisibilityrelations in Theorem 2.11. For polynomial ideals of dimension 0, their W-characteristic sets do notinvolve any parameters and thus the condition is satisfied naturally. For the rest of the paper, weassume that the condition is also satisfied for the positive-dimensional case where the structuresof lex Gr¨obner bases are rather complicated. Decomposition into characteristic pairs
In this section we discuss the decomposition of an arbitrary polynomial set into (strong) charac-teristic pairs with associated zero relations and prove some properties about the decomposition.A decomposition algorithm will be presented in Section 4.
Definition 3.1
A pair ( G , C ) with G , C ⊆ K [ x ] is called a characteristic pair in K [ x ] if G is areduced lex Gr¨obner basis, C is the W-characteristic set of hGi , and C is normal.The following known results (Propositions 3.2–3.4) concerning normal sets are recalled, withreferences or self-contained proofs, exhibiting some of the nice properties of characteristic pairs. Proposition 3.2
For any zero-dimensional normal set
N ⊆ K [ x ] : ( a ) sat( N ) = hN i ; ( b ) N isthe lex Gr¨obner basis of hN i . Proof As N is a zero-dimensional normal set, each ini( N ) is a constant in K for N ∈ N . Thenstatement (a) follows directly from the definition of sat( N ), and statement (b) can be derivedeasily by using [12, Section 2.9, Thm. 3 and Prop. 4]. (cid:3) Proposition 3.3 ([33, Prop. 2.2])
Let
N ⊆ K [ x ] be any positive-dimensional normal set withparameters ˜ x ⊆ x . Then N is the lex Gr¨obner basis of hN i over K ( ˜ x ) . Let T = [ T , . . . , T r ] ⊆ K [ x ] be an arbitrary triangular set with parameters x , . . . , x d ( d + r = n ). For each i = 0 , . . . , r −
1, write T ≤ i = T ∩ K [ x , . . . , x d + i ] = [ T , . . . , T i ] , I ≤ i = ini( T ) ∩ K [ x , . . . , x d + i ] . T is said to have the projection property if for any i = 0, . . . , r − x i ∈ Z ( T ≤ i / I ≤ i ), thereexist ¯ x i +1 , . . . , ¯ x r ∈ ¯ K such that ( ¯ x i , ¯ x i +1 , . . . , ¯ x r ) ∈ Z ( T / ini( T )). Here empty T ≤ i and I ≤ i areunderstood as { } and { } respectively. Proposition 3.4
Any normal set
N ⊂ K [ x ] has the projection property. Proof
Let N = [ N , . . . , N r ], and I i = ini( N i ) for i = 1 , . . . , r . Since N is a normal set, I ≤ i = { I , . . . , I r } for i = 0 , . . . , r . Thus for any i = 0 , . . . , r − x i ∈ Z ( N ≤ i / I ≤ i ), I j ( ¯ x i ) = 0 forall j = i + 1 , . . . , r , so there exist ¯ x i +1 , . . . , ¯ x r ∈ ¯ K such that ( ¯ x i , ¯ x i +1 , . . . , ¯ x r ) ∈ Z ( N / ini( N )). (cid:3) Remark 3.5
In general regular sets do not have the projection property. Consider, for example, T = [ x − u, xy + 1] ⊆ Q [ u, x, y ], where Q is the field of rational numbers and u < x < y . Then u is the parameter of T . Now T ≤ = I ≤ = ∅ and the parametric value ¯ u = 0 ∈ ¯ Q = Z ( T ≤ / I ≤ ),but Z ( T / ini( T )) = ∅ when u = ¯ u . Proposition 3.6
Let C be the normal W-characteristic set of hPi ⊆ K [ x ] . If sat( C ) = hCi , then sat( C ) = hPi . Proof
The proposition follows immediately from hCi ⊆ hPi ⊆ sat( C ) (Proposition 2.10(b)). (cid:3) The reverse direction of Proposition 3.6, namely sat( C ) = hPi implies sat( C ) = hCi , is notcorrect in general. For example, G = { y , xz + y, yz, z } ⊆ K [ x, y, z ] is a reduced lex Gr¨obner basiswith x < y < z : the normal W-characteristic set of hGi is C = [ y , xz + y ], and one can check that hGi = sat( C ), but hCi 6 = sat( C ).What is of special interest between G and C in a characteristic pair ( G , C ) is whether the equality hGi = sat( C ) holds. This equality does hold when sat( C ) = hCi (according to Proposition 3.6), butthe condition sat( C ) = hCi does not necessarily hold as the above example shows. Moreover, it iscomputationally difficult to verify whether sat( T ) = hT i holds for a triangular set T [1, 30]. efinition 3.7 A characteristic pair ( G , C ) is said to be strong if sat( C ) = hGi . Definition 3.8
A reduced lex Gr¨obner basis G is said to be characterizable if hGi = sat( C ), where C is the W-characteristic set of G .It is easy to see that every W-characteristic set is determined by a reduced lex Gr¨obner basis,while a characterizable Gr¨obner basis is also determined by its W-characteristic set. A strongcharacteristic pair thus furnishes a characterizable Gr¨obner basis with a normal W-characteristicset. In what follows we show that the W-characteristic set of any characterizable Gr¨obner basisis normal, so that the characterizable Gr¨obner basis and its W-characteristic set form a strongcharacteristic pair. Proposition 3.9
The W-characteristic set of any characterizable Gr¨obner basis is normal.
Proof
We prove the proposition by contradiction. Suppose that the W-characteristic C = [ C , . . . , C r ]of the characterizable Gr¨obner basis G is abnormal. Then by Theorem 2.11 there exist two polyno-mials C k +1 and C l ( l ≤ k ) in C with lv( C l ) = lv( I k +1 ) such that either (a) prem(ini( C k +1 ) , [ C , . . . , C l ])) =0, when ini( C k +1 ) is not R-reduced with respect to C l ; or (b) prem( C l , [ C , . . . , C l − , ini( C k +1 )]) =0, when ini( C k +1 ) is R-reduced.Let I i = ini( C i ) for i = 1 , . . . , l and I k +1 = ini( C k +1 ).For case (a), from the pseudo-remainder formula we know that I k +1 ∈ sat( C ). Write C k +1 = I k +1 lv( C k +1 ) d + R , where deg( R, lv( C k +1 )) < d . If R = 0, then lv( C k +1 ) d ∈ sat( C ), butlv( C k +1 ) d < lex C k +1 , which contradicts with the minimality of G as the reduced lex Gr¨obnerbasis of sat( C ); If R = 0, clearly R ∈ sat( C ), but R
6∈ hGi for R is B-reduced with respect to G ,which contradicts the equality hGi = sat( C ).For case (b), it follows from the pseudo-remainder formula that there exist i , . . . , i l ∈ Z ≥ (the set of nonnegative integers) and Q , . . . , Q l ∈ K [ x ] such that I i · · · I i l − l − ini( I k +1 ) i l C l = Q C + · · · + Q l I k +1 ;clearly Q l ∈ sat( C ). Since deg( I k +1 , lv( C l )) < deg( C l , lv( C l )) in this case and all I , . . . , I l − involve only the parameters, we have lv( Q l ) = lv( C l ) but deg( Q l , lv( C l )) < deg( C l , lv( C l )), andthus Q l < lex C l . This contradicts with the minimality of G . (cid:3) Remark 3.10
Gr¨obner bases are good representations of polynomial ideals. Here it is shown thatthe W-characteristic set C of a characterizable Gr¨obner basis G provides another representation ofthe same ideal hGi . The representation C is simpler than G because C is a subset of G , whereas G can be computed from C if needed. In other words, characterizable Gr¨obner bases are those specialGr¨obner bases whose W-characteristic sets can characterize or represent the ideals they generate. Let F be a finite, nonempty set of polynomials in K [ x ]. We call a finite set { ( G , C ) , . . . , ( G t , C t ) } of characteristic pairs in K [ x ] a characteristic decomposition of F if the following zero relationshold: Z ( F ) = t [ i =1 Z ( G i ) = t [ i =1 Z ( C i / ini( C i )) = t [ i =1 Z (sat( C i )) . (1) Theorem 3.11
From any finite, nonempty polynomial set
F ⊆ K [ x ] , one can compute in a finitenumber of steps a characteristic decomposition of F . The above theorem will be proved by giving a concrete algorithm (Algorithm 1 in Section 4.1)with correctness and termination proof (in Section 4.2). In what follows, we focus our attentionon the properties of characteristic decomposition. emark 3.12 From any characteristic decomposition of a polynomial set F , one can extract anormal decomposition {C , . . . , C t } of F , with each C i a normal set for i = 1 , . . . , t and Z ( F ) = S ti =1 Z ( C i / ini( C i )). The projection property of normal sets (see Proposition 3.4) allows us towrite down the conditions on the parameters for a normal set to have zeros for the variables,which makes normal decomposition an appropriate approach for parametric polynomial systemsolving [8, 28, 20]. The obtained parametric conditions are not necessarily disjoint and thus do notnecessarily lead to a partition of the parameter space. However, since the conditions derived fromnormal sets are expressed by means of initials which involve only the parameters, it is easier tocompute comprehensive triangular decompositions [8] via normal decomposition than via regulardecomposition.A proper ideal in K [ x ] is said to be purely equidimensional if its associated prime ideals are allof the same height. According to [2, Prop. 4.1.3], sat( T ) is purely equidimensional for any regularset T ⊆ K [ x ]. Thus it follows from Proposition 3.6 that for any characteristic pair ( G , C ) in thecharacteristic decomposition of F , hGi is purely equidimensional if sat( C ) = hCi is verified. Moreprecisely, we have the following. Proposition 3.13
Let Ψ be a characteristic decomposition of F ⊆ K [ x ] and assume that sat( C ) = hCi for every characteristic pair ( G , C ) ∈ Ψ . Then p hFi = T ( G , C ) ∈ Ψ p hGi and each hGi is purelyequidimensional. In fact, the ideal hGi in Proposition 3.13 is also strongly equidimensional according to [2,Thm. 4.1.4]. The following theorem shows how a Ritt characteristic set of a polynomial ideal canbe constructed from the W-characteristic set (when it is normal) of the ideal.
Theorem 3.14
Let C = [ C , . . . , C r ] be the W-characteristic set of hPi ⊆ K [ x ] and C ∗ =[ C , prem( C , [ C ]) , . . . , prem( C r , [ C ,. . ., C r − ])] . (2) If C is normal, then the following statements hold: ( a ) C ∗ is a normal set; ( b ) C ∗ is a Ritt characteristic set of hPi ; ( c ) Z ( C ∗ / ini( C ∗ )) = Z ( C / ini( C )) . Proof
Let C ∗ = [ C ∗ , . . . , C ∗ r ], I i = ini( C i ), and I ∗ i = ini( C ∗ i ) for i = 1 , . . . , r .(a–b) According to [43, Thm. 3.4], C ∗ is a regular set and C ∗ is a Ritt characteristic set of hPi ; hence it suffices to prove that C ∗ is normal. Since C ∗ i = prem( C i , [ C , . . . , C i − ]) for any i = 1 , . . . , r , there exist q , . . . , q i − ∈ Z ≥ and Q , . . . , Q i − ∈ K [ x ] such that I q · · · I q i − i − C i = Q C + · · · + Q i − C i − + C ∗ i . (3)Since ini( C i ) does not involve any of lv( C ) , . . . , lv( C i − ), we have ini( C ∗ i ) = I q · · · I q i − i − I i for i = 1 , . . . , r ; thus C ∗ is normal.(c) On one hand, for any ¯ x ∈ Z ( C / ini( C )), C i ( ¯ x ) = 0 and I i ( ¯ x ) = 0 for i = 1 , . . . , r . From(3) we know that C ∗ i ( ¯ x ) = 0 and I ∗ i ( ¯ x ) = 0 for i = 1 , . . . , r ; or equivalently ¯ x ∈ Z ( C ∗ / ini( C ∗ )).On the other hand, for any ˆ x ∈ Z ( C ∗ / ini( C ∗ )), C ∗ i ( ˆ x ) = 0 and I ∗ i ( ˆ x ) = 0 for i = 1 , . . . , r . Clearly C = C ∗ , C ( ˆ x ) = 0 and I ( ˆ x ) = 0. Suppose now that C i ( ˆ x ) = 0 and I i ( ˆ x ) = 0 hold for i = 2 , . . . , k −
1. Then by (3) and I ∗ k = I q · · · I q k − k − I k we have C k ( ˆ x ) = 0 and I k ( ˆ x ) = 0. Byinduction, ˆ x ∈ Z ( C / ini( C )). (cid:3) Corollary 3.15
Let Ψ be a characteristic decomposition of F ⊆ K [ x ] and C ∗ be computed from C according to (2) for each characteristic pair ( G , C ) ∈ Ψ . Then Z ( F ) = [ ( G , C ) ∈ Ψ Z ( C ∗ / ini( C ∗ )) = [ ( G , C ) ∈ Ψ Z (sat( C ∗ )) , and C ∗ is the Ritt characteristic set of hGi for each ( G , C ) ∈ Ψ . orollary 3.16 Let P , C , and C ∗ be as in Theorem 3.14 and ˜ x ⊆ x be the parameters of C . Thenboth C and C ∗ are lex Gr¨obner bases of hPi over K ( ˜ x ) . Furthermore, let C ∗ = [ C ∗ , . . . , C ∗ r ] . Then [ C ∗ ini( C ∗ ) − , . . . , C ∗ r ini( C ∗ r ) − ] is the reduced lex Gr¨obner basis of hPi over K ( ˜ x ) . Proof
By Proposition 3.3, C is the lex Gr¨obner basis of hCi over K ( ˜ x ). As sat( C ) = hCi over K ( ˜ x ), from Proposition 3.6 we know that C is the lex Gr¨obner basis of hPi over K ( ˜ x ). Moreover,since each ini( C ∗ i ) is a constant in K ( ˜ x ) for i = 1 , . . . , r and deg( C ∗ j , lv( C ∗ i )) < deg( C ∗ i , lv( C ∗ i ))(1 ≤ i, j ≤ r , j = i ), [ C ∗ ini( C ∗ ) − , . . . , C ∗ r ini( C ∗ r ) − ] is the reduced lex Gr¨obner basis of hPi over K ( ˜ x ). (cid:3) Note that the R-reduced normal set C ∗ computed in (2) is not necessarily a subset of G (whilethe W-characteristic set C is). Corollary 3.16 does not hold when the W-characteristic set C is abnormal. In fact, the problem of constructing a Ritt characteristic set of an ideal I fromthe reduced lex Gr¨obner basis of I was already studied by Aubry, Moreno Maza, and Lazard intheir influential paper [3] of 1999. It is the first author of this paper who pointed out with anexample in [43] that the relevant results including Theorem 3.1, Proposition 3.4, and Theorem 3.2in Section 3 of [3] are flawed. He showed that the construction is simple and straightforward inthe normal case and made the problem of effective construction open again for the abnormal case.The incorrectness of the construction process in [3] is caused essentially by the non-closeness ofaddition on polynomials having pseudo-remainder 0 with respect to an irregular triangular set T ,that is, prem( P, T ) = prem( Q, T ) = 0 does not necessarily imply prem( P + Q, T ) = 0 for arbitrarypolynomials P and Q . The non-closeness seems to be a major obstacle for constructing abnormalRitt characteristic sets of polynomial ideals. Now we show that for any characteristic decomposition Ψ = { ( G , C ) , . . . , ( G t , C t ) } of F ⊆ K [ x ],one can explicitly transform Ψ into a strong characteristic decomposition ¯Ψ = { ( ¯ G , ¯ C ) , . . . , ( ¯ G t , ¯ C t ) } ,where each ( ¯ G i , ¯ C i ) is a strong characteristic pair for i = 1 , . . . , t . Lemma 3.17 ([43, Lem. 2.4])
Let T = [ T , . . . , T r ] ⊆ K [ x ] be a regular set with lv( T r ) < x n ,and P = P d x dm + · · · + P x m + P ∈ K [ x ] be a polynomial with lv( P ) = x m > lv( T r ) and deg( P, x m ) = d . Then prem( P, T ) = 0 if and only if prem( P i , T ) = 0 for all i = 0 , , . . . , d . Lemma 3.18 ([40, Lem. 6.2.6])
Let T be a regular set in K [ x ] . Then for any F ∈ K [ x ] , if res( F, T ) = 0 , then sat( T ) : F ∞ = sat( T ) . Theorem 3.19
Let ( G , C ) be a characteristic pair, ¯ G be the reduced lex Gr¨obner basis of sat( C ) ,and ¯ C be the W-characteristic set of h ¯ Gi . Then the following statements hold: ( a ) the parameters of ¯ C coincide with those of C ; ( b ) ¯ C is a normal set, sat( ¯ C ) = sat( C ) , and thus ( ¯ G , ¯ C ) is a strong characteristic pair. Proof (a) Let lv( G ) = { lv( G ) : G ∈ G} and lv( ¯ G ) = { lv( ¯ G ) : ¯ G ∈ ¯ G} . It suffices to prove thatlv( G ) = lv( ¯ G ).(lv( G ) ⊇ lv( ¯ G )) Suppose that x k ∈ lv( ¯ G ), but x k lv( G ). Then there exists a ¯ G ∈ ¯ G withlv( ¯ G ) = x k . Since ¯ G ∈ h ¯ Gi = sat( C ), prem( ¯ G, C ) = 0. Write ¯ G = ¯ H p x pk + · · · + ¯ H , where p = deg( ¯ G ) and ¯ H i ∈ K [ x , . . . , x k − ] for i = 0 , . . . , p . Since x k lv( G ) and prem( ¯ G, C ) = 0,from Lemma 3.17 we know that for i = 0 , . . . , p , prem( ¯ H i , C ) = 0 and thus ¯ H i ∈ sat( C ) = h ¯ Gi .This implies that nform( ¯ H i , ¯ G \ { ¯ G } ) = 0 for i = 0 , . . . , p , and thus nform( ¯ G, ¯ G \ { ¯ G } ) = 0, whichcontradicts with ¯ G being the reduced lex Gr¨obner basis.(lv( G ) ⊆ lv( ¯ G )) Suppose that x l ∈ lv( G ), but x l lv( ¯ G ). Then there exists a C ∈ C ⊆ G withlv( C ) = x l , and thus C ∈ sat( C ) = h ¯ Gi ; it follows that nform( C, ¯ G ) = 0. Write C = ini( C ) x dl + R ,where d = deg( C, x l ) and R ∈ K [ x , . . . , x l ]. Then from nform( C, ¯ G ) = 0 and x l lv( ¯ G ), we knowthat nform(ini( C ) , ¯ G ) = 0, and thus ini( C ) ∈ h ¯ Gi = sat( C ). It follows that prem(ini( C ) , C ) = 0;this contradicts with the fact that C ∈ C and C is a normal set. b) Let C = [ C , . . . , C r ]. By (a) we can assume that ¯ C = [ ¯ C , . . . , ¯ C r ] with lv( ¯ C i ) = lv( C i ) for i = 1 , . . . , r . Since ¯ C i ∈ h ¯ Gi = sat( C ) and C is regular, prem( ¯ C i , C ) = 0; since C i ∈ sat( C ) = h ¯ Gi and¯ C is the W-characteristic set of ¯ G , prem( C i , ¯ C ) = 0. This leads to deg( ¯ C i , lv( ¯ C i )) = deg( C i , lv( C i )).We first prove that ¯ C is a normal set, namely ¯ I i := ini( ¯ C i ) only involves the parameters foreach i = 1 , . . . , r . If, otherwise, some ¯ I i involves the variables in lv( ¯ G ), then, under the assumptionthat all the parameters are ordered smaller than the variables in lv( ¯ G ), we have C i < lex ¯ C i , fordeg( ¯ C i , lv( ¯ C i )) = deg( C i , lv( C i )). But C i ∈ h ¯ Gi , and this contradicts with the minimality of ¯ C i asan element in ¯ G .Next we show the equality sat( ¯ C ) = sat( C ). Since h ¯ Gi = sat( C ), it suffices to show that h ¯ Gi = sat( ¯ C ). On one hand, from Proposition 2.10(b) we know that h ¯ Gi ⊆ sat( ¯ C ). On the otherhand, let ¯ I = Q ¯ C ∈ ¯ C ini( ¯ C ). Since ¯ I only involves the parameters of C , res( ¯ I, C ) = ¯ I = 0. ByLemma 3.18 we have sat( C ) : ¯ I ∞ = h ¯ Gi . With the inclusion ¯ C ⊆ ¯ G , the following relationsat( ¯ C ) = h ¯ Ci : ¯ I ∞ ⊆ h ¯ Gi : ¯ I ∞ = sat( C ) : ¯ I ∞ = h ¯ Gi holds. (cid:3) Example 3.20
The pair( G , C ) = ( { y , x z + xy, yz + xz + y } , [ y , x z + xy ])is a characteristic pair in Q [ x, y, z ] with x < y < z . The reduced lex Gr¨obner basis ¯ G of sat( C )is { y , xz + y, yz, z } , so the W-characteristic set ¯ C of ¯ G is [ y , xz + y ]. One can check that theparameter of both C and ¯ C is x , ¯ C is normal, and sat( C ) = sat( ¯ C ). Lemma 3.21
Let C = [ C , . . . , C r ] and ¯ C = [ ¯ C , . . . , ¯ C r ] be as in Theorem 3.19. Then ini( ¯ C i ) | ini( C i ) for i = 1 , . . . , r . Proof
From the proof of Theorem 3.19 we know that prem( ¯ C i , C ) = 0, prem( C i , ¯ C ) = 0, anddeg( ¯ C i , lv( ¯ C i )) = deg( C i , lv( C i )) for i = 1 , . . . , r . Thenini( C i ) ¯ C i = ini( ¯ C i ) C i mod sat( C ≤ i − ) , where C ≤ i − = [ C , . . . , C i − ]. Therefore, ini( ¯ C i ) | ini( C i ) ¯ C i modulo sat( C ≤ i − ).If ini( ¯ C i ) | ini( C i ) modulo sat( C ≤ i − ), then clearly ini( ¯ C i ) | ini( C i ), for both ini( ¯ C i ) and ini( C i )involve only the parameters. Otherwise, there exists an ¯ I K such that ¯ I | ini( ¯ C i ) and ¯ I | ¯ C i modulo sat( C ≤ i − ). Then by ( ¯ C i / ¯ I ) ¯ I = ¯ C i ∈ h ¯ Ci , we have ¯ C i / ¯ I ∈ sat( ¯ C ) = h ¯ Gi , but ¯ C i / ¯ I < lex ¯ C i ,which contradicts with the minimality of ¯ C i as a polynomial in the W-characteristic set. (cid:3) Theorem 3.22
Let
Ψ = { ( G , C ) , . . . , ( G t , C t ) } be a characteristic decomposition of F ⊆ K [ x ] .For each ( G i , C i ) ∈ Ψ , let ( ¯ G i , ¯ C i ) be the corresponding strong characteristic pair as constructed inTheorem 3.19, i = 1 , . . . , t . Then Z ( F ) = t [ i =1 Z ( ¯ G i ) = t [ i =1 Z ( ¯ C i / ini( ¯ C i )) = t [ i =1 Z (sat( ¯ C i )) . Proof
Note that the zero relation (1) holds for the characteristic decomposition Ψ, which maybe computed according to Theorem 3.11. The first and the third equality above can be proved byusing the equalities sat( C i ) = sat( ¯ C i ) in Theorem 3.19(b) and h ¯ G i i = sat( C i ): Z ( F ) = t [ i =1 Z (sat( C i )) = t [ i =1 Z (sat( ¯ C i )) = t [ i =1 Z ( ¯ G i ) . Now we prove that Z ( F ) = S ti =1 Z ( ¯ C i / ini( ¯ C i )). Since Z ( ¯ C i / ini( ¯ C i )) ⊆ Z (sat( ¯ C i )), we have t [ i =1 Z ( ¯ C i / ini( ¯ C i )) ⊆ t [ i =1 Z (sat( ¯ C i )) = Z ( F ) . o show the other inclusion, observe first that Z ( C i / ini( C i )) ⊆ Z (sat( C i )) = Z ( ¯ G i ) ⊆ Z ( ¯ C i ) . (4)for i = 1 , . . . , t . Write C i = [ C i , . . . , C ir i ] and ¯ C i = [ ¯ C i , . . . , ¯ C ir i ]. Then for any ¯ x ∈ Z ( C i / ini( C i )), Q r i j =1 ini( C ij )( ¯ x ) = 0. By Lemma 3.21 we have ini( ¯ C ij ) | ini( C ij ) for j = 1 , . . . , r i , and thus Q r i j =1 ini( ¯ C ij )( ¯ x ) = 0. Combining this inequality with (4), we have Z ( C i / ini( C i )) ⊆ Z ( ¯ C i / ini( ¯ C i )),and thus Z ( F ) = S ti =1 Z ( C i / ini( C i )) ⊆ S ti =1 Z ( ¯ C i / ini( ¯ C i )). This completes the proof. (cid:3) Remark 3.23
As shown by Theorems 3.19 and 3.22, any characteristic decomposition Ψ of apolynomial set can be transformed into a strong characteristic decomposition of the polynomialset by computing the reduced lex Gr¨obner basis of the saturated ideal of the W-characteristic setin each characteristic pair in Ψ. No splitting occurs in this transformation: the number of strongcharacteristic pairs produced by the transformation is the same as that of the characteristic pairsin Ψ.
In this section we present an algorithm that computes a characteristic decomposition of any finite,nonempty set of nonzero polynomials.
An overall strategy based on Theorem 2.11 for characteristic decomposition is sketched in [43,Sect. 4]. Following this strategy, we detail the decomposition method below.Let Φ be a set of polynomial sets, initialized as {F} with
F ⊆ K [ x ] being the input set, andΨ be a set of characteristic pairs already computed. Now we pick a polynomial set P ∈
Φ andremove it from Φ, compute the reduced lex Gr¨obner basis G of the ideal hPi , and extract theW-characteristic set C = [ C , . . . , C r ] of hPi from G . Let I i = ini( C i ) for i = 1 , . . . , r .1. If C is normal, then from Proposition 2.10(c) we know that Z ( C / ini( C )) ⊆ Z ( P ) ⊆ Z ( C ) . (5)In view of this zero relation, we put the characteristic pair ( G , C ) into Ψ and adjoin thepolynomial sets G ∪ { I } , . . . , G ∪ { I r } to Φ for further processing.2. If C is not normal, then by Theorem 2.11 we have certain pseudo-divisibility relations betweenpolynomials in C and can use them to split G as follows, where the integers k and l are as inTheorem 2.11.2.1. If I k +1 is not R-reduced with respect to C l , then the polynomial sets G ∪ { I } , . . . , G ∪{ I l } , G ∪ { I k +1 } are adjoined to Φ.2.2. If I k +1 is R-reduced with respect to C l , then let Q = pquo( C l , I k +1 ) be the pseudo-quotient of C l with respect to I k +1 and C l − = [ C , . . . , C l − ].2.2.1. If prem(ini( Q ) , C l − ) = 0, then the polynomial sets G ∪ { I } , . . . , G ∪ { I l − } , G ∪{ ini( I k +1 ) } are adjoined to Φ.2.2.2. If prem(ini( Q ) , C l − ) = 0, then the polynomial sets G ∪ { I } , . . . , G ∪ { I l − } , G ∪{ prem( Q, C l − ) } , G ∪ { I k +1 } are adjoined to Φ.After the splitting of G , we continue picking another polynomial set P ′ (and meanwhile removeit) from Φ, compute the reduced lex Gr¨obner basis G ′ of hP ′ i , extract the W-characteristic set of hP ′ i from G ′ , and split G ′ when necessary. This process is repeated until Φ becomes empty.The method for characteristic decomposition, whose main steps are outlined above, is describedformally as Algorithm 1. Algorithm 1
Ψ =
CharDec ( F ). Given a finite, nonempty set F of nonzero polynomials in K [ x ],this algorithm computes a characteristic decomposition Ψ of F , or the empty set meaning that Z ( F ) = ∅ .
1. Set Ψ = ∅ and Φ = {F} .C2. Repeat the following steps until Φ = ∅ :C2.1. Pick P ∈
Φ and remove it from Φ.C2.2. Compute the reduced lex Gr¨obner basis G of hPi .C2.3. If G = { } , then go to C2; otherwise:C2.3.1. Extract the W-characteristic set C from G .C2.3.2. If C is normal (case 1), then reset Ψ with Ψ ∪ { ( G , C ) } , and Φ withΦ ∪ {G ∪ { ini( C ) } | ini( C ) K , C ∈ C} , (6)and go to C2.C2.3.3. Pick the smallest polynomial C in C such that [ T ∈ C | lv( T ) ≤ lv( C )] as atriangular set is abnormal.C2.3.4. Let I = ini( C ) and y = lv( I ).C2.3.5. Pick the polynomial C ∗ in C such that lv( C ∗ ) = y .C2.3.6. If I is not R-reduced with respect to C ∗ (case 2.1), then reset Φ withΦ ∪ {G ∪ { ini( T ) } | lv( T ) ≤ y, T ∈ C} ∪ {G ∪ { I }} , (7)and go to C2.C2.3.7. If prem(ini( Q ) , [ T ∈ C | lv( T ) < y ]) = 0 (case 2.2.1), then reset Φ withΦ ∪ {G ∪ { ini( T ) } | lv( T ) < y, T ∈ C} ∪ {G ∪ { ini( I ) }} , (8)and go to C2.C2.3.8. Reset Φ with Φ ∪ {G ∪ { ini( T ) | lv( T ) < y, T ∈ C}} ∪{G ∪ { prem( Q, [ T ∈ C | lv( T ) < y ]) } , G ∪ { I }} (9)(case 2.2.2) and go to C2.C3. Output Ψ. Theorem 4.1
Algorithm 1 terminates in a finite number of steps with correct output.
Proof ( Termination ) The process of splitting in Algorithm 1 can be viewed as building up atree from its root as the input set F . Every time a polynomial set P is picked from Φ, splittingoccurs according to one of the four cases, treated in steps C2.3.2, C2.3.6, C2.3.7, and C2.3.8 ofAlgorithm 1, as long as the reduced lex Gr¨obner basis of hPi is not { } (for otherwise P has nozero). Suppose that the split polynomial sets G , . . . , G s are adjoined to Φ. In the sense of buildingup the tree, this means that the child nodes of P are G , . . . , G s .To prove the termination of Algorithm 1, we need to show that each path in the tree is of finitelength. Thus by the Ascending Chain Condition (see, e.g., [12, Chap. 2, Thm. 7]), it suffices toshow that for all the four cases of splitting, each polynomial set G ′ = G ∪ { H } adjoined to Φ forsome H generates an ideal hG ′ i that is strictly greater than hGi , or equivalently H
6∈ hGi .Let the W-characteristic set C in step C2.3.1 be written as [ C , . . . , C r ] with I i = ini( C i ) and C i = [ C , . . . , C i ] for 1 ≤ i ≤ r . Then C , I , and C ∗ in steps C2.3.3–C2.3.5 correspond to C k +1 , I k +1 ,and C l respectively for some integers l ≤ k as stated in Theorem 2.11.Let H K be the initial of some C ∈ C as in (6) in step C2.3.2. We claim that H is B-reducedwith respect to G ; for otherwise C will be reducible by some polynomial in G \ { C } , which conflictswith the fact that G is the reduced lex Gr¨obner basis of hPi . Therefore, H
6∈ hGi for all C ∈ C in(6). With similar arguments, one can show that H
6∈ hGi as well if H K is the initial of somepolynomial T ∈ C as in (7), (8), and (9).To complete the proof of termination, it remains to show that H = prem( Q, C l − )
6∈ hGi , where Q = pquo( C l , I k +1 ). In step C2.3.8, the conditions lv( Q ) > lv( C l − ) and prem(ini( Q ) , C l − ) = 0 old. By Theorem 2.11, C l − is normal and thus regular; then by Lemma 3.17, prem( Q, C l − ) = 0and deg(prem( Q, C l − ) , lv( C i )) < deg( C i , lv( C i )) (10)for i = 1 , . . . , l −
1. Furthermore, as Q = pquo( C l , I k +1 ) and lv( C l ) also appears in I k +1 (byTheorem 2.11), we have deg( Q, lv( C l )) < deg( C l , lv( C l )) and thusdeg(prem( Q, C l − ) , lv( C l )) < deg( C l , lv( C l )) . (11)Since C is the W-characteristic set of G , the relations (10) and (11) imply that prem( Q, C l − ) isB-reduced with respect to G and thus prem( Q, C l − )
6∈ hGi .( Correctness ) When Z ( F ) = ∅ , G = { } in step C2.3 and Ψ = ∅ is returned. Therefore, toprove the correctness of Algorithm 1, we need to show that when Z ( F ) = ∅ , Ψ is a characteristicdecomposition of F , namely all the pairs ( G , C ) ∈ Ψ are characteristic pairs and the zero relation(1) holds.It is clear that each ( G , C ) ∈ Ψ is a characteristic pair, for only in step C2.3.2 is the outputset Ψ adjoined with a new pair ( G , C ), where G is a reduced lex Gr¨obner basis and C is its normalW-characteristic set. We first prove that Z ( F ) = S ( G , C ) ∈ Ψ Z ( C / ini( C )) by considering all the fourcases of splitting. For this purpose, let C = [ C , . . . , C r ] be the W-characteristic set, I i = ini( C i )(1 ≤ i ≤ r ), and L = ini( I k +1 ).Case 1 (step C2.3.2): the W-characteristic set C is normal. In this case, let J = I · · · I r . Then Z ( P ) = ( Z ( P ) \ Z ( J )) ∪ Z ( P ∪ { J } ). It follows from the zero relation (5) that Z ( P ) \ Z ( J ) = Z ( C / ini( C )). Moreover, as J = 0 implies that I = 0, or I = 0 , . . . , or I r = 0, Z ( P ∪ { J } ) = r [ i =1 Z ( P ∪ { I i } ) = r [ i =1 Z ( G ∪ { I i } ) . Therefore, Z ( P ) = Z ( C / ini( C )) ∪ r [ i =1 Z ( G ∪ { I i } ) . (12)Case 2.1 (step C2.3.6): C is abnormal and deg( I k +1 , lv( I k +1 )) ≥ deg( C l , lv( I k +1 )). By Theo-rem 2.11(a) we have prem( I k +1 , C l ) = 0, and thus there exist Q , . . . , Q l ∈ K [ x ] and q , . . . , q l ∈ Z ≥ such that I q · · · I q l l I k +1 = Q C + · · · + Q l C l . This means that I q · · · I q l l I k +1 ∈ h C , . . . , C l i ⊆hPi = hGi , so Z ( P ) = Z ( G ) = Z ( G ∪ { I q · · · I q l l I k +1 } ) = l [ i =1 Z ( G ∪ { I i } ) ∪ Z ( G ∪ { I k +1 } ) . (13)Case 2.2.1 (step C2.3.7): C is abnormal, deg( I k +1 , lv( I k +1 )) < deg( C l , lv( I k +1 )), and prem(ini( Q ) , C l − ) = 0. By the formula L q C l = QI k +1 + R of pseudo-division of C l with respect to I k +1 , where q ∈ Z ≥ and Q, R ∈ K [ x ], we have ini( Q ) = L q − I l . Sinceprem(ini( Q ) , C l − ) = prem( L q − I l , C l − ) = I l prem( L q − , C l − ) = 0 , prem( L q − , C l − ) = 0 and thus Z ( P ) = l − [ i =1 Z ( G ∪ { I i } ) ∪ Z ( G ∪ { L } ) (14)follows from arguments similar to those in case 2.1.Case 2.2.2 (step C2.3.8): C is abnormal, deg( I k +1 , lv( I k +1 )) < deg( C l , lv( I k +1 )), and prem(ini( Q ) , C l − ) = 0. By Theorem 2.11(b), we haveprem( C l , [ C , . . . , C l − , I k +1 ]) = prem(prem( C l , I k +1 ) , C l − )= prem( L q C l − QI k +1 , C l − ) = 0 , here Q and q are as in case 3. Then there exist q , . . . , q l − ∈ Z ≥ and Q , . . . , Q l − ∈ K [ x ] suchthat I q · · · I q l − l − ( L q C l − QI k +1 ) = P l − i =1 Q i C i . It follows that − I q · · · I q l − l − I k +1 Q = l X i =1 Q i C i (15)for Q l = − I q · · · I q l − l − L q . Let the formula of pseudo-division of Q with respect to C l − be I ¯ q · · · I ¯ q l − l − Q = l − X i =1 ¯ Q i C i + prem( Q, C l − ) (16)for some ¯ q , . . . , ¯ q l − ∈ Z ≥ and ¯ Q , . . . , ¯ Q l − ∈ K [ x ]. Then one can find ˆ q , . . . , ˆ q l − ∈ Z ≥ andˆ Q , . . . , ˆ Q l ∈ K [ x ] such that I ˆ q · · · I ˆ q l − l − I k +1 prem( Q, C l − ) = l X i =1 ˆ Q i C i holds (in view of the formulas (15) and (16)). Therefore, Z ( P ) = l − [ i =1 Z ( G ∪ { I i } ) ∪ Z ( G ∪ { I k +1 } ) ∪ Z ( G ∪ { prem( Q, C l − ) } ) . (17)The zero relations in (12), (13), (14), and (17) show that for each polynomial set P ∈
Φ, anyzero of P is either in Z ( C / ini( C )) if the W-characteristic set C of hPi is normal in case 1 or in Z ( P ′ ) for another polynomial set P ′ adjoined to Φ for later computation in the other cases. Thisproves the zero relation Z ( F ) = S ( G , C ) ∈ Ψ Z ( C / ini( C )); for Algorithm 1 terminates when Φ becomesempty.On one hand, by the zero relation (5), we have Z ( F ) = [ ( G , C ) ∈ Ψ Z ( C / ini( C )) ⊆ [ ( G , C ) ∈ Ψ Z ( G ) . On the other hand, Z ( G ) ⊆ Z ( F ) holds for all ( G , C ) ∈ Ψ according to the zero relations (12), (13),(14), and (17) for the four cases of splitting. This proves the equality Z ( F ) = S ( G , C ) ∈ Ψ Z ( G ). Foreach ( G , C ) ∈ Ψ, we have Z ( C / ini( C )) ⊆ Z (sat( C )) ⊆ Z ( G ), and thus Z ( F ) = [ ( G , C ) ∈ Ψ Z ( C / ini( C )) ⊆ [ ( G , C ) ∈ Ψ Z (sat( C )) ⊆ [ ( G , C ) ∈ Ψ Z ( G ) = Z ( F ) . This completes the proof of the zero relation (1). (cid:3)
Let F = { ay − x − , − xyz + az, xz − az + y } ⊆ K [ a, x, y, z ] with a < x < y < z . The procedureto compute a characteristic decomposition of F using Algorithm 1 is shown in Table 1, where G i is the computed reduced lex Gr¨obner basis and C i is its W-characteristic set in the i th loop.The polynomial sets F i and polynomials G j in Table 1 are listed below: F = { x + 1 , y, ay, az, ( y + a ) z, G } , F = { G , aG , zG , G , G , G } , F = { a, x + 1 , y, z } , F = { G , G , G , G } , F = { a, x + 1 , x + x, G , xyz, G } , F = { a, x + 1 , x + x, xy, G } ; G = x + 2 x + (1 − a ) x − a , G = ay − x − ,G = x y + xy − ax − a, G = xy − x − ,G = x + x − a , G = xy − a,G = z − yz + y − y. The output characteristic decomposition of P , as shown in Table 1, is { ( G , C ) , ( G , C ), ( G , C ) , ( G , C ) } . i Φ P G i C i Normal Case Ψ1 {F} F { G , G , G , G , zG , zG , G } [ G , G , zG ] No 2.2.2 ∅ {F , F } F { x + 1 , y, az, z } [ x + 1 , y, az ] Yes 1 { ( G , C ) } {F , F } F { G , G , G ,zG , G } [ G , G , zG ] No 2.1 { ( G , C ) } {F , F , F } F { a, x + 1 , y, z } [ a, x + 1 , y, z ] Yes 1 { ( G , C ) , ( G , C ) } {F , F } F { G , G , G , G } [ G , G , G ] Yes 1 { ( G , C ), ( G , C )( G , C ) } {F , F } F { a, x + 1 , y , yz, z − y } [ a, x + 1 , y , yz ] No 2.2.2 ( {G , C ), ( G , C )( G , C ) } {F } F { a, x + 1 , y, z } [ a, x + 1 , y, z ] Yes 1 { ( G , C ), ( G , C )( G , C ), ( G , C ) } Algorithm 1 has been implemented in
Maple
17 based on functions available in the
FGb and
Maple ’s built-in packages for Gr¨obner basis computation. The implementation will be includedin the upcoming new version of the Epsilon package for triangular decomposition [41].In Theorem 2.11 there is an assumption on the variable order (i.e., all the parameters of aW-characteristic set are ordered before the other variables). This assumption always holds inthe zero-dimensional case. Our experiments show that in the positive-dimensional case there areabout one fourth of the test examples for which it happens that the assumption does not hold.The assumption is made to ensure that the (pseudo-) divisibility relationships in Theorem 2.11occur. In the case where the assumption does not hold, we can make such relationships to occurby changing the variable order heuristically. In fact, using the heuristics we were able to obtainnecessary (pseudo-) divisibility relationships to complete the characteristic decomposition for allthe test examples.Let us emphasize that Algorithm 1 decomposes any polynomial set into characteristic pairsof reduced lex Gr¨obner bases and their W-characteristic sets at one stroke. The two kinds ofobjects resulted from the combined decomposition, each having its own structures and properties,are interconnected. This makes our algorithm distinct from other existing ones for triangulardecomposition. To observe the computational performance of the algorithm, in comparison withalgorithms for indirect normal decomposition (that is, first computing a regular decompositionand then normalizing the regular sets in the decomposition), we made some experiments on anIntel(R) Core(TM) Quad CPU at 2.83 GHz with 4.00 GB RAM under Windows 7 Home Basic.Selected results of the experiments are presented in Table 2, of which the first 9 are taken from[8] and the others are from benchmarks for the
FGb library. We implemented Algorithm 1 as
CharDec in Maple for characteristic decomposition and used the functions
Triangularize (from the
RegularChains package in
Maple ) and
RegSer (from the
Epsilon package for
Maple ) for regulardecomposition and the function normat (from the miscel module of
Epsilon ) for normalization.In Table 2, “Source” indicates the label in the above-cited references and “Dim” denotes thedimension of the ideal in the example. “Total” under
CharDec records the total time (followed bythe number of pairs in parenthesis) for characteristic decomposition using Algorithm 1; “GB” under
CharDec records the time for computing all the reduced lex Gr¨obner bases; “Total” and “Regular”under
RegSer and
Triangularize record the total time for normal decomposition and the time forregular decomposition (followed by the numbers of components in parenthesis) respectively, wherenormal decompositions are computed from regular decompositions by means of normalization using normat . The marks “lost” and “ > CharDec is for the computation of lex Gr¨obner bases, as one
CharDec RegSer Triangularize
Source Dim Total GB Total Regular Total RegularS5 4 0.14(8) 0.047 4.182(31) 0.484(19) 1.513(9) 0.124(1)S7 1 0.156(5) 0.078 0.251(7) 0.11(4) 0.249(5) 0.109(1)S8 2 0.062(2) 0.32 0.062(3) 0.062(3) 0.156(2) 0.141(2)S9 2 0.125(5) 0.078 0.483(21) 0.14(8) 0.188(6) 0.094(1)S10 3 0.594(16) 0.313 0.438(7) 0.172(5) 0.36(3) 0.235(1)S13 3 0.312(13) 0.14 0.171(8) 0.109(8) 0.125(2) 0.094(1)S14 2 0.531(9) 0.327 0.14(6) 0.109(6) 0.157(8) 0.125(8)S16 3 0.640(6) 0.344 0.703(7) 0.609(7) 4.609(8) 4.609(8)S17 6 lost lost lost lost lost lostnueral 1 1.826(15) 1.514 > > > > > > > > > > > > can see from Table 2. In our implementation, the FGb library is first invoked to compute Gr¨obnerbases with respect to graded reverse lexicographic term ordering, and the computed Gr¨obner basesare then converted to lex Gr¨obner bases by changing the term ordering using either the FGLMalgorithm for the zero-dimensional case [18] or the Gr¨obner walk otherwise [11]. Unfortunately,the built-in implementation of the Gr¨obner walk algorithm in
Maple is very inefficient and it isthe current bottleneck of our implementation.Finally, we add a few remarks to conclude the paper: we have studied characteristic pairs (that is,pairs of reduced lex Gr¨obner bases and normal triangular sets) and the problem of characteristicdecomposition (that is, decomposition of arbitrary polynomial sets into characteristic pairs). Wehave proved a number of properties about characteristic pairs and characteristic decomposition,and proposed an algorithm with implementation for the decomposition. The algorithm exploresthe inherent connection between Ritt characteristic sets and lex Gr¨obner bases and involves mainlythe computation of lex Gr¨obner bases; normal triangular sets are obtained as by-product almostfor free. Associated to a characteristic decomposition { ( G , C ) , . . . , ( G t , C t ) } of a polynomial set P are zero decompositions Z ( P ) = Z ( G ) ∪ · · · ∪ Z ( G t ) = Z ( C / ini( C t )) ∪ · · · ∪ Z ( C t / ini( C t ))and the corresponding radical ideal decompositions p hPi = p hG i ∪ · · · ∪ p hG t i = p sat( C ) ∪ · · · ∪ p sat( C t ) . In these decompositions, the reduced lex Gr¨obner bases G , . . . , G t and normal triangular sets C , . . . , C t are closely linked and well structured polynomial sets whose usefulness has been widelyrecognized. References [1] G. Angerm¨uller. Triangular systems and a generalization of primitive polynomials.
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