Improved Computation of Involutive Bases
aa r X i v : . [ c s . S C ] M a y Improved Computation of Involutive Bases
Bentolhoda Binaei , Amir Hashemi , , and Werner M. Seiler Department of Mathematical Sciences, Isfahan University of TechnologyIsfahan, 84156-83111, Iran; School of Mathematics, Institute for Research in Fundamental Sciences (IPM),Tehran, 19395-5746, Iran [email protected]@cc.iut.ac.ir Institut f¨ur Mathematik, Universit¨at KasselHeinrich-Plett-Straße 40, 34132 Kassel, Germany [email protected]
Abstract.
In this paper, we describe improved algorithms to computeJanet and Pommaret bases. To this end, based on the method proposedby M¨oller et al. [21], we present a more efficient variant of Gerdt’s al-gorithm (than the algorithm presented in [17]) to compute minimal in-volutive bases. Further, by using the involutive version of Hilbert driventechnique, along with the new variant of Gerdt’s algorithm, we modifythe algorithm, given in [24], to compute a linear change of coordinates fora given homogeneous ideal so that the new ideal (after performing thischange) possesses a finite Pommaret basis. All the proposed algorithmshave been implemented in
Maple and their efficiency is discussed via aset of benchmark polynomials.
Gr¨obner bases are one of the most important concepts in computer algebra fordealing with multivariate polynomials. A Gr¨obner basis is a special kind of gen-erating set for an ideal which provides a computational framework to determinemany properties of the ideal. The notion of Gr¨obner bases was originally in-troduced in 1965 by Buchberger in his Ph.D. thesis and he also gave the basicalgorithm to compute it [2, 3]. Later on, he proposed two criteria for detectingsuperfluous reductions to improve his algorithm [1]. In 1983, Lazard [20] devel-oped new approach by making connection between Gr¨obner bases and linearalgebra. In 1988, Gebauer and M¨oller [10] reformulated Buchberger’s criteria inan efficient way to improve Buchberger’s algorithm. Furthermore, M¨oller et al. in[21] proposed an improved version of Buchberger’s algorithm by using the syzy-gies of constructed polynomials to detect useless reductions (this algorithm maybe considered as the first signature-based algorithm to compute Gr¨obner bases).Relying on the properties of the Hilbert series of an ideal, Traverso [27] describedthe so-called
Hilbert-driven Gr¨obner basis algorithm to improve Buchberger’s al-gorithm by discarding useless critical pairs. In 1999, Faug`ere [6] presented hisF algorithm to compute Gr¨obner bases which stems from Lazard’s approach20] and uses fast linear algebra techniques on sparse matrices (this algorithmhas been efficiently implemented in Maple and
Magma ). In 2002, Faug`ere pre-sented the famous F algorithm for computing Gr¨obner bases [7]. The efficiencyof this signature-based algorithm benefits from an incremental structure andtwo new criteria, namely F and IsRewritten criteria (nowadays known respec-tively as signature and syzygy criteria). We remark that several authors havestudied signature-based algorithms to compute Gr¨obner bases and as the novelapproaches in this directions we refer to e.g. [8, 9]. Involutive bases may be considered as an extension of Gr¨obner bases (w.r.t.a restricted monomial division) for polynomial ideals which include additionalcombinatorial properties. The origin of involutive bases theory must be tracedback to the work of Janet [19] on a constructive approach to the analysis oflinear and certain quasi-linear systems of partial differential equations. ThenJanet’s approach was generalized to arbitrary (polynomial) differential systemsby Thomas [26]. Based on the related methods developed by Pommaret in hisbook [22], the notion of involutive polynomial bases was introduced by Zharkovand Blinkov in [28]. Gerdt and Blinkov [13] introduced a more general conceptof involutive division and involutive bases for polynomial ideals, along with al-gorithmic methods for their construction. An efficient algorithm was devised byGerdt [12] (see also [16]) for computing involutive and Gr¨obner bases using theinvolutive form of Buchberger’s criteria (see http://invo.jinr.ru for theefficiency analysis of the implementation of this algorithm). In this paper, werefer to this algorithm as
Gerdt’s algorithm . Finally, Gerdt et al. [17] describeda signature-based algorithm (with an incremental structure) to apply the F criterion for deletion of unnecessary reductions. Some of the drawbacks of thisalgorithm are as follows: Due to its incremental structure (in order to apply theF criterion), the selection strategy should be the POT module monomial or-dering (which may be not efficient in general). Further, to respect the signatureof computed polynomials, the reduction process may be not accomplished and(that may increase the number of intermediate polynomials) that may signifi-cantly affect the efficiency of computation. Finally, the involutive basis that thisalgorithm returns may be not minimal.The aim of this paper is to provide an effective method to calculate Pommaretbases . These bases introduced by Zharkov and Blinkov in [28] are a particularform of involutive bases containing many combinatorial properties of the idealsthey generate, see e.g. [23–25] for a comprehensive study of Pommaret bases.They are not only of interest in computational aspects of algebraic geometry(e.g. by providing deterministic approaches to transform a given ideal into someclasses of generic positions [24]), but they also serve in theoretical aspects ofalgebraic geometry (e.g. by providing simple and explicit formulas to read offmany invariants of an ideal like dimension, depth and Castelnuovo-Mumfordregularity [24]).Relying on the method developed by M¨oller et al. [21], we give a new signature-based variant of Gerdt’s algorithm to compute minimal involutive bases. In par-ticular, the experiments show that the new algorithm is more efficient than Gerdtt al. algorithm [17]. On the other hand, [24] proposes an algorithm to compute deterministically a linear change of coordinates for a given homogeneous idealso that the changed ideal (after performing this change) possesses a finite Pom-maret basis (note that in general a given ideal does not have a finite Pommaretbasis). In doing so, one computes iteratively the Janet bases of certain polyno-mial ideals. By applying the involutive version of Hilbert driven technique on thenew variant of Gerdt’s algorithm, we modify this algorithm to compute Pom-maret bases. We have implemented all the algorithms described in this articleand we assess their performance on a number of test examples.The rest of the paper is organized as follows. In the next section, we willreview the basic definitions and notations which will be used throughout thispaper. Section 3 is devoted to the description of the new variant of Gerdt’salgorithm. In Section 4, we present the improved algorithm to compute a linearchange of coordinates for a given homogeneous ideal so that the new ideal has afinite Pommaret basis. We analyze the performance of the proposed algorithmsin Section 5. Finally, in Section 6 we conclude the paper by highlighting theadvantages of this work and discussing future research directions.
In this section, we review the basic definitions and notations from the theory ofGr¨obner bases and involutive bases that will be used in the rest of the paper.Throughout this paper we assume that P = k [ x , . . . , x n ] is the polynomialring (where k is an infinite field). We consider also homogeneous polynomials f , . . . , f k ∈ P and the ideal I = h f , . . . , f k i generated by them. We denotethe total degree of and the degree w.r.t. a variable x i of a polynomial f ∈ P respectively by deg( f ) and deg i ( f ). Let M = { x α · · · x α n n | α i ≥ , ≤ i ≤ n } be the monoid of all monomials in P . A monomial ordering on M is denoted by ≺ and throughout this paper we shall assume that x n ≺ · · · ≺ x . The leadingmonomial of a given polynomial f ∈ P w.r.t. ≺ will be denoted by LM( f ). If F ⊂ P is a finite set of polynomials, we denote by LM( F ) the set { LM( f ) | f ∈ F } . The leading coefficient of f , denoted by LC( f ), is the coefficient ofLM( f ). The leading term of f is defined to be LT( f ) = LM( f ) LC( f ). A finiteset G = { g , . . . , g k } ⊂ P is called a Gr¨obner basis of I w.r.t ≺ if LM( I ) = h LM( g ) , . . . , LM( g k ) i where LM( I ) = h LM( f ) | f ∈ Ii . We refer e.g. to [4] formore details on Gr¨obner bases.Let us recall the definition of Hilbert function and Hilbert series of a homo-geneous ideal. Let X ⊂ P and s a positive integer. We define the degree s part X s of X to be the set of all homogeneous elements of X of degree s . Definition 1.
The
Hilbert function of I is defined by HF I ( s ) = dim k ( P s / I s ) where the right-hand side denotes the dimension of P s / I s as a k -linear space. It is well-known that the Hilbert function of I is the same as that of LT( I ) (seee.g. [4, Prop. 4, page 458]) and therefore the set of monomials not contained inLT( I ) forms a basis for P s / I s as a k -linear space (Macaulay’s theorem). Thisbservation is the key idea behind the Hilbert-driven Gr¨obner basis algorithm.Roughly speaking, suppose that I is a homogeneous ideal and we want to com-pute a Gr¨obner basis of I by Buchberger’s algorithm in increasing order w.r.t.the total degree of the S-polynomials. Assume that we know beforehand HF I ( s )for a positive integer s . Suppose that we are at the stage where we are looking atthe critical pairs of degree s . Consider the set P of all critical pairs of degree s .Then, we compare HF I ( s ) with the Hilbert function at s of the ideal generatedby the leading terms of all already computed polynomials. If they are equal, wecan remove P .Below, we review some definitions and relevant results on involutive basestheory (see [12] for more details). We recall first involutive divisions based onpartitioning the variables into two subsets of the variables, the so-called multi-plicative and non-multiplicative variables . Definition 2. An involutive division L is given on M if for any finite set U ⊂ M and any u ∈ U , the set of variables is partitioned into the subset ofmultiplicative M L ( u, U ) and non-multiplicative variables N M L ( u, U ) such thatthe following three conditions hold where L ( u, U ) denotes the monoid generatedby M L ( u, U ) :1. v, u ∈ U , u L ( u, U ) ∩ v L ( v, U ) = ∅ ⇒ u ∈ v L ( v, U ) or v ∈ u L ( u, U ) ,2. v ∈ U , v ∈ u L ( u, U ) ⇒ L ( v, U ) ⊂ L ( u, U ) ,3. V ⊂ U and u ∈ V ⇒ L ( u, U ) ⊂ L ( u, V ) .We shall write u | L w if w ∈ u L ( u, U ) . In this case, u is called an L -involutivedivisor of w and w an L -involutive multiple of u . We recall the definitions of the Janet and the Pommaret division, respectively.
Example 3.
Let U ⊂ P be a finite set of monomials. For each sequence d , . . . , d n of non-negative integers and for each 1 ≤ i ≤ n we define the subsets[ d , . . . , d i ] = { u ∈ U | d j = deg j ( u ) , ≤ j ≤ i } . The variable x is Janet multiplicative (denoted by J -multiplicative) for u ∈ U if deg ( u ) = max { deg ( v ) | v ∈ U } . For i > x i is Janet multi-plicative for u ∈ [ d , . . . , d i − ] if deg i ( u ) = max { deg i ( v ) | v ∈ [ d , . . . , d i − ] } . Example 4.
For u = x d · · · x d k k with d k > { x k , . . . , x n } are con-sidered as Pommaret multiplicative (denoted by P -multiplicative) and the othervariables as Pommaret non-multiplicative. For u = 1 all the variables are multi-plicative. The integer k is called the class of u and is denoted by cls( u ).The Pommaret division is called a global division , because the assignment of themultiplicative variables is independent of the set U . In order to avoid repeatingnotations let L always denote an involutive division. Definition 5.
The set F ⊂ P is called involutively head autoreduced if for each f ∈ F there is no h ∈ F \ { f } with LM( h ) | L LM( f ) . efinition 6. Let I ⊂ P be an ideal. An L -involutively head autoreduced subset G ⊂ I is an L -involutive basis for I (or simply either an involutive basis or L -basis) if for all f ∈ I there exists g ∈ G so that LM( g ) | L LM( f ) .Example 7. Let I = { x x , x x , x x } ⊂ k [ x , x , x ]. Then, { x x , x x , x x ,x x } is a Janet basis for I and { x x , x x , x x , x x , x i +31 x , x i +31 x | i ≥ } isa (infinite) Pommaret basis for I . Indeed, Janet division is Noetherian, howeverPommaret division is non-Noetherian (see [14] for more details).Gerdt in [12] proposed an efficient algorithm to construct involutive bases basedon a completion process where prolongations of generators by non-multiplicativevariables are reduced. This process terminates in finitely many steps for anyNoetherian division. Definition 8.
Let F ⊂ P be a finite. Following the notations in [24], the invo-lutive span generated by F is denoted by h F i L , ≺ . Thus, a set F ⊂ I is an involutive basis for I if we have I = h F i L , ≺ . Definition 9.
Let F ⊂ I be an involutively head autoreduced set of homoge-neous polynomials. The involutive Hilbert function of F is defined by IHF F ( s ) =dim k ( P s / ( h F i L , ≺ ) s ) . Since F is involutively head autoreduced, one easily recognizes that h F i L , ≺ = L f ∈ F k [ M L (LM( f ) , LM( F ))] · f . Thus using the well-known combinatorial for-mulas to count the number of monomials in certain variables, we getIHF I ( s ) = (cid:18) n + s − s (cid:19) − X f ∈ F (cid:18) s − deg( f ) + k f − s − deg( f ) (cid:19) where k f is the number of multiplicative variables of f (see e.g. [12]). We remarkthat an involutively head autoreduced subset F ⊂ I is an involutive basis for I if and only if HF I ( s ) = IHF F ( s ) for each s . We now propose a variant of Gerdt’s algorithm [12] by using the intermediatecomputed syzygies to compute involutive bases and especially Janet bases. Forthis, we recall briefly the signature-based variant of M¨oller et al. algorithm [21]to compute Gr¨obner bases (the practical results are given in Section 5).
Definition 10.
Let us consider F = ( f , . . . , f k ) ∈ P k . The (first) syzygy mod-ule of F is defined to be Syz( F ) = { ( h , . . . , h k ) | h i ∈ P , P ki =1 h i f i = 0 } . Schreyer in his master thesis proposed a slight modification of Buchberger’salgorithm to compute a Gr¨obner basis for the module of syzygies of a Gr¨obnerbasis. The construction of this basis relies on the following key observation (see[5]): Let G = { g , . . . , g s } be a Gr¨obner basis. By tracing the dependency of eachPoly( g i , g i ) on G we can write SPoly( g i , g j ) = P sk =1 a ijk g k with a ijk ∈ P . Let e , . . . , e s be the standard basis for P s and m ij = lcm (LT( g i ) , LT( g j )). Set s ij = m i,j / LT( g i ) . e i − m i,j / LT( g j ) . e j − ( a ij e + a ij e + · · · + a ijs e s ) . Definition 11.
Let G = { g , . . . , g s } ⊂ P . Schreyer’s module ordering is definedas follows: x β e j ≺ s x α e i if LT( x β g j ) ≺ LT( x α g i ) and breaks ties by i < j . Theorem 12 (Schreyer’s Theorem).
For a Gr¨obner basis G = { g , . . . , g s } the set { s ij | ≤ i < j ≤ s } forms a Gr¨obner basis for Syz( g , . . . , g s ) w.r.t. ≺ s .Example 13. Let F = { xy − x, x − y } ⊂ k [ x, y ]. The Gr¨obner basis of F w.r.t. x ≺ dlex y is G = { g = xy − x, g = x − y, g = y − y } and the Gr¨obner basisof Syz( g , g , g ) is { ( x, − y + 1 , − , ( − x, y − , − x + y + 1) , ( y, , − x ) } .According to this observation, M¨oller et al. [21] proposed a variant of Buch-berger’s algorithm by using the syzygies of constructed polynomials to removesuperfluous reductions. Algorithm 1 below corresponds to it with a slight mod-ification to derive a signature-based algorithm to compute Gr¨obner bases. Weassociate to each polynomial f , the two-tuple p = ( f, m e i ) where Poly( p ) = f isthe polynomial part of f and Sig( p ) = m e i is its signature. Further, the function NormalForm ( f, G ) returns a remainder of the division of f by G . Further, ifSig( p ) = m e i in the first step of reduction process we must not use f i ∈ G . We Algorithm 1
Gr¨obnerBasis
Input:
A set of polynomials F ⊂ P ; a monomial ordering ≺ Output:
A Gr¨obner basis G for h F i G := {} and syz := {} P := { ( F [ i ] , e i ) | i = 1 , . . . , | F |} while P = ∅ do select (using normal strategy) and remove p ∈ P if ∄ s ∈ syz s.t. s | Sig( p ) then f := Poly( p ) h := NormalForm ( f, G ) syz := syz ∪ { Sig( p ) } if h = 0 then j := | G | + 1 for g ∈ G do P := P ∪ { ( r.h, r. e j ) } s.t. r. LM( h ) = LCM(LM( g ) , LM( h )) G := G ∪ { h } syz := syz ∪ { LM( g ) . e j | LM( h ) and LM( g ) are coprime } end forend ifend ifend whilereturn ( G ) show now how to apply this structure to improve Gerdt’s algorithm [16]. efinition 14. Let F = ( f , . . . , f k ) ⊂ P k be a sequence of polynomials. The involutive syzygy module ISYZ( F ) of F is the set of all ( h , . . . , h k ) ∈ P k sothat P ki =1 h i f i = 0 where h i ∈ k [ M L (LM( f i ) , LM( F ))] . [24, Thm. 5.10] contains an involutive version of Schreyer’s theorem replacingS-polynomials by non-multiplicative prolongations and using involutive division.Algorithm 2 below represents the new variant of Gerdt’s algorithm for comput-ing involutive bases using involutive syzygies. For this purpose, we associate toeach polynomial f , the quadruple p = ( f, g, V, m. e i ) where f = Poly( p ) is thepolynomial itself, g = Anc( p ) is its ancestor, V = NM( p ) is the list of non-multiplicative variables of f which have been already processed in the algorithmand m. e i = Sig( p ) is the signature of f . If P is a set of quadruple, we denote byPoly( P ) the set { Poly( p ) | p ∈ P } . Algorithm 2
InvolutiveBasis
Input:
A finite set F ⊂ P ; an involutive division L ; a monomial ordering ≺ Output:
A minimal L -basis for h F i F :=sort( F, ≺ ) T := { ( F [1] , F [1] , ∅ , e ) } Q := { ( F [ i ] , F [ i ] , ∅ , e i ) | i = 2 , . . . , | F |} syz := {} while Q = ∅ do Q :=sort( Q, ≺ s ) p := Q [1] if ∄ s ∈ syz s.t s | Sig( p ) with non-constant quotient then h := InvolutiveNormalForm ( p, T, L , ≺ ) syz := syz ∪ { h [2] } if h = 0 and LM(Poly( p )) = LM(Anc( p )) then Q := { q ∈ Q | Anc( q ) = Poly( p ) } end ifif h = 0 and LM(Poly( p )) = LM( h ) thenfor q ∈ T with proper conventional division LM(Poly( h )) | LM(Poly( q )) do Q := Q ∪ { q } T := T \ { q } end for j := | T | + 1 T := T ∪ { ( h, h, ∅ , e j ) } else T := T ∪ { ( h, Anc( p ) , NM( p ) , Sig( p )) } end iffor q ∈ T and x ∈ NM L (LM(Poly( q )) , LM(Poly( T )) \ NM( q )) do Q := Q ∪ { ( x. Poly( q ) , Anc( q ) , ∅ , x. Sig( q )) } NM( q ) := NM( q ) ∪ NM L (LM(Poly( q )) , LM(Poly( T ))) ∪ { x } end forend ifend whilereturn (Poly( T )) n this algorithm, the functions sort( X, ≺ ) and sort( X, ≺ s ) sort X by increas-ing, respectively, LM( X ) w.r.t. ≺ and { Sig( p ) | p ∈ X } w.r.t. ≺ s . The involutivenormal form algorithm is given in Algorithm 3. Algorithm 3
InvolutiveNormalForm
Input:
A quadruple p ; a set of quadruples T ; an involutive division L ; a monomialordering ≺ Output: An L -normal form of p modulo T , and the corresponding signature, if any S := {} and h := Poly( p ) and G := Poly( T ) while h has a monomial m which is L -divisible by G do select g ∈ G with LM( g ) | L m if m = LM(Poly( p )) and ( m/ LM( g ) . Sig( g ) = Sig( p ) or Criteria ( h, g )) thenreturn (0 , S ) end ifif m = LM(Poly( p )) and m/ LM( g ) . Sig( g ) ≺ s Sig( p ) then S := S ∪ { Sig( p ) } end if h := h − cm/ LT( g ) .g where c is the coefficient of m in h end whilereturn ( h, S ) Furthermore, we apply the involutive form of Buchberger’s criteria from [12].We say that
Criteria ( p, g ) holds if either C ( p, g ) or C ( p, g ) holds where C ( p, g ) is true if LM(Anc( p )) . LM(Anc( g )) = LM(Poly( p )) and C ( p, g ) is trueif LCM(LM(Anc( p )) , LM(Anc( g ))) properly divides LM(Poly( p )). Remark 15.
We shall remark that, due to the second if -loop in Algorithm 3,if m i e i is added into syz then there exists an involutive representation of theform m i g i = P ℓj =1 h j g j + h where T = { g , . . . , g ℓ } ⊂ P is the output of thealgorithm, h is L -normal form of p modulo T and LM( h j ) e j ≺ s m i e i for each j .In the next proof, by an abuse of notation, we refer to the signature of aquadruple as the signature of its polynomial part. Theorem 16.
InvolutiveBasis terminates in finitely many steps (if L is aNoetherian division) and returns a minimal involutive basis for its input ideal.Proof. The termination and correctness of the algorithm are inherited from thoseof Gerdt’s algorithm [12] provided that we show that any polynomial removedusing syzygies is superfluous. This happens in both algorithms. Let us dealfirst with Algorithm 2. Now, suppose that for p ∈ Q there exists s ∈ syz so that s | Sig( p ) with non-constant quotient. Suppose that Sig( p ) = m i e i and s = m ′ i e i where m i = um ′ i with u = 1. Let T = { g , . . . , g ℓ } ⊂ P bethe output of the algorithm and m ′ i g i = P ℓj =1 h j g j + h be the representa-tion of m ′ i g i with g j ∈ T, h, h j ∈ P and h the involutive remainder of thedivision of m ′ i g i by T . Then, from the structure of both algorithms, it yieldshat LM( h j g j ) ≺ LM( m ′ i g i ). In particular, we have LM( h j ) e j ≺ s m ′ i e i for each j . This follows that LM( uh j ) e j ≺ s um ′ i e i = m i e i for each j . On the otherhand, if h = 0 then again by the structure of the algorithm uh has a signa-ture less than m i e i . For each j and for each term t in h j we know that thesignature of utg j is less than m i e i and by the selection strategy used in thealgorithm which is based on Schreyer’s ordering, utg j should be studied before m ′ i g i and therefore it has an involutive representation in terms of T . Further-more, the same holds also for uh provided that h = 0. These arguments showthat m ′ i g i is unnecessary and it can be omitted. Now we turn to Algorithm 3.Let p ∈ Q and g ∈ T so that LM( h ) = u LT( g ) and Sig( p ) = u Sig( g ) where h = Poly( p ) and u is a monomial. Using the above notations, let Sig( p ) = m i e i and Sig( g ) = m ′ i e i where m i = um ′ i . Further, let m ′ i g i = P ℓj =1 h j g j + g bethe representation of m ′ i g i with LM( h j ) e j ≺ s m ′ i e i for each j . It follows fromthe assumption that LM( h j g j ) ≺ LM( m ′ i g i ) = LM( g ) for each j . We can write um ′ i g i = P ℓj =1 uh j g j + ug . Since LM( uh j ) e j ≺ s um ′ i e i = m i e i for each j then,by repeating the above argument, we deduce that uh j g j for each j has an invo-lutive representtaion. Therefore, um ′ i g i has a representation using the fact that u is multiplicative for g . Thus h has a representation and it can be removed. ⊓⊔ As we mentioned Pommaret division is not Noetherian and therefore, a givenideal may not have a finite Pommaret basis. However, if the ideal is in quasi-stable position (see Def. 19) it has a finite Pommaret basis. On the other hand,a generic linear change of variables transforms an ideal in such a position. Thus,one of the challenges in this direction is to find a linear change of variables sothat the ideal after performing this change possesses a finite Pommaret basis.[24] proposes a deterministic algorithm to compute such a linear change by com-puting repeatedly the Janet basis of the last transformed ideal. In this section,by using the algorithm described in Section 3, we show how one can incorporatean involutive version of Hilbert driven strategy to improve this algorithm.
Algorithm 4
HDQuasiStable
Input:
A finite set F ⊂ P and a monomial ordering ≺ Output:
A linear change Φ so that h Φ ( F ) i has a finite Pommaret basis Φ := ∅ and J := InvolutiveBasis ( F, J , ≺ ) and A := test (LM( J ) , ≺ ) while A = true do G := substitution of φ := A [3] A [3] + cA [2] in J for a random choice of c ∈ KT emp := HDInvolutiveBasis ( G, J , ≺ ) B := test (LM( T emp )) if B = A then Φ := Φ, φ and J := T emp and A := B end ifend whilereturn ( Φ ) t is worth noting that in [24] it is proposed to perform a Pommaret headautoreduced process on the calculated Janet basis at each iteration. However,we do not need to perform this operation because each computed Janet basisis minimal and by [11, Cor. 15] each minimal Janet basis is Pommaret headautoreduced. All the used functions are described below. By the structure of thealgorithm, we first compute a Janet basis for the input ideal using Involutive-Basis algorithm. From this basis, one can read off easily the Hilbert functionof the input ideal. Further, the Hilbert function of an ideal does not changeafter performing a linear change of variables. Thus we can apply this Hilbertfunction in the next Janet bases computations as follows. The algorithm has thesame structure as the
InvolutiveBasis algorithm and so we remove the similarlines. We add the next written lines in
HDInvolutiveBasis algorithm between p := Q [1] and the first if -loop in InvolutiveBasis algorithm.
Algorithm 5
HDInvolutiveBasis
Input:
A set of monomials F ; an involutive division L ; a monomial ordering ≺ Output:
A minimal L -involutive basis for h F i ... d := deg( p ) while HF h F i ( d ) = IHF T ( d ) do remove from Q all q ∈ Q s.t. deg(Poly( q )) = d if Q = ∅ thenreturn (Poly( T )) else p := Q [1] d := deg( p ) end ifend while ... Algorithm 6 test
Input:
A finite set U of monomials Output:
True if any element of U has the same number of Pommaret and Janetmultiplicative variables, and false otherwise if ∃ u ∈ U s.t. M P , ≺ ( u, U ) = M J , ≺ ( u, U ) then V := M J , ≺ ( u, F ) \ M P , ≺ ( u, F ) return ( false, V [1] , x cls( u ) ) end ifreturn ( true ) Theorem 17.
HDQuasiStable algorithm terminates in finitely many stepsand it returns a linear change of variables for a given homogeneous ideal so thatthe changed ideal (after performing the change on the input ideal) possesses afinite Pommaret basis.roof.
Let I be the ideal generated by F ; the input of HDQuasiStable al-gorithm. The termination of this algorithm follows, from one side, from thetermination of the algorithms to compute Janet bases. From the other side, [24,Prop. 2.9] shows that there exists an open Zariski set U of k n × n so that foreach linear change of variables, say Φ corresponding to an element of U we have Φ ( I ) has a finite Pommaret basis. Moreover, he proved that the process of find-ing such a linear change termintaes in finitely many steps (see [24, Rem. 9.11]).Taken together, these arguments show that HDQuasiStable algorithm termi-nates. To prove the correctness, using the notations of
HDInvolutiveBasis algorithm, we shall prove that any p ∈ Q removed by Hilbert driven strategyreduces to zero. In this direction, we recall that any change of variables is alinear automorphism of P , [18, page 52]. Thus, for each i , the dimension over k of components of degree i of I and that of I after the change remains stable.This yields that the Hilbert function of I does not change after a linear changeof variables. Let J be the Janet basis computed by InvolutiveBasis . One canreadily observe that HF I ( d ) = IHF J ( d ) for each d , and therefore from the firstJanet basis one can derive the Hilbert function of I and use it to improve thenext Janet bases computations. Now, suppose that F is the input of HDInvo-lutiveBasis algorithm, p ∈ Q and HF I ( d ) = IHF T ( d ) for d = deg(Poly( p )). Itfollows that dim k ( h F i d ) = dim k ( h Poly( T ) i d ) and therefore the polynomials ofPoly( T ) generate involutively whole h F i d and this shows that p is superfluouswhich ends the proof. ⊓⊔ Remark 18.
We remark that we assumed that the input of
InvolutiveBasis and
HDQuasiStable algorithms should be homogeneous, however the formeralgorithm works also for non-homogeneous ideals as well. Further, the latter algo-rithm also may be applied for non-homogeneous ideals provided that we considerthe affine Hilbert function for such ideals; i.e. HF I ( s ) = dim k ( P ≤ s / I ≤ s ).[24] provides a number of equivalent characterizations of the ideals which havefinite Pommaret bases. Indeed, a given ideal has a finite Pommaret basis if onlyif the ideal is in quasi stable position (or equivalently if the coordinates are δ -regular) see [24, Prop. 4.4]. Definition 19.
A monomial ideal I is called quasi stable if for any monomial m ∈ I and all integers i, j, s with ≤ j < i ≤ n and s > , if x si | m there existsan integer t ≥ such that x tj m/x si ∈ I . A homogeneous ideal I is in quasi stableposition if LT( I ) is quasi stable.Example 20. The ideal I = h x x , x , x i ⊂ k [ x, y, z ] is a quasi stable monomialideal and its Pommaret basis is { x x , x , x , x x x , x x , x x x , x x } . We have implemented both algorithms
InvolutiveBasis and
HDQuasiStable in Maple 17 . It is worth noting that, in the given paper, we are willing to The
Maple code of the implementations of our algorithms and examples are avail-able at http://amirhashemi.iut.ac.ir/softwares ompare behavior of
InvolutiveBasis and
HDQuasiStable algorithms withGerdt et al. [17] and
QuasiStable [24] algorithms, respectively (we shall remarkthat
QuasiStable has the same structure as the
HDQuasiStable , however tocompute Janet bases we use Gerdt’s algorithm). For this purpose, we used somewell-known examples from computer algebra literature. All computations weredone over Q , and for the input degree-reverse-lexicographical monomial order-ing. The results are shown in the following tables where the time and memorycolumns indicate, respectively, the consumed CPU time in seconds and amountof megabytes of used memory. The C and C columns show, respectively, thenumber of polynomials removed by C and C criteria by the correspondingalgorithm. The sixth column shows the number of polynomials eliminated bythe new criterion related to syzygies applied in InvolutiveBasis and
Invo-lutiveNormalForm algorithms. The F and S columns show the number ofpolynomials removed, respectively, by F and super-top-reduction criteria. Threelast columns represent, respectively, the number of reductions to zero, the num-ber and the maximum degree of polynomials in the final involutive basis (wenote that for Gerdt et al. algorithm the number of polynomials is the size of thebasis after the minimal process). The computations in this paper are performedon a personal computer with 2 .
70 GHz Intel(R) Core(TM) i7-2620M CPU, 8 GBof RAM, 64 bits under the Windows 7 operating system.
Liu time memory C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis C C Syz F S redz poly deg
InvolutiveBasis
As one can observe
InvolutiveBasis is a signature-based variant of Gerdt’salgorithm which has a structure closer to Gerdt’s algorithm and it is more effi-cient than Gerdt et al. algorithm. Moreover, we can see the detection of criteriand the number of reductions to zero by the algorithms are different. Indeed,this difference is due to the selection strategy used in each algorithm. More pre-cisely, in the Gerdt et al. algorithm the set of non-multiplicative prolongations issorted by POT ordering however in
InvolutiveBasis it is sorted using Schreyerordering. However, one needs to implement it efficiently in C/C++ to be ableto compare it with GINV software .The next tables illustrate an experimental comparison of HDQuasiStable and
QuasiStable algorithms. In these tables HD column shows the number ofpolynomials removed by Hilbert driven strategy in the corresponding algorithm.Further, the chen column shows the number of linear changes that one needsto transform the corresponding ideal into quasi stable position. The deg columnrepresents the maximum degree of the output Pommaret basis (which is theCastelnuovo-Mumford regularity of the ideal, see [24]). Further, each columnshows the number of detection by corresponding criterion for all computed Janetbases. Finally, we shall remark that in the next tables we use the homogenizationof the generating set of the test examples used in the previous tables. In addition,the computation of Janet basis of an ideal generated by a set F and the one ofthe ideal generated by the homogenization of F are not the same. For example,the CPU time to compute the Janet basis of the homogenization of Lichtblauexample is 270 .
24 sec..
Liu time memory C C Syz HD redz chen deg
HDQuasiStable
QuasiStable C C Syz HD redz chen deg
HDQuasiStable
QuasiStable C C Syz HD reds chen deg
HDQuasiStable
QuasiStable C C Syz HD redz chen deg
HDQuasiStable
QuasiStable C C Syz HD redz chen deg
HDQuasiStable
QuasiStable C C Syz HD redz chen deg
QuasiStable
QuasiStable C C Syz HD redz chen deg
HDQuasiStable
QuasiStable C C Syz HD redz chen deg
HDQuasiStable
QuasiStable See http://invo.jinr.ru
Conclusion and Perspective
In this paper, a modification of Gerdt’s algorithm [16] which is a signature-basedversion of the involutive algorithm [12, 16] to compute minimal involutive basesis suggested. Additionally, we present a Hilbert driven optimization of the pro-posed algorithm, to compute (finite) Pommaret bases. In doing so, the proposedalgorithm computes iteratively Janet bases by using the modified Gerdt’s al-gorithm and use them, in accordance to ideas of [24], to perform the variabletransformations. The new algorithms have been implemented in
Maple andthey are compared with the Gerdt’s algorithm and with the algorithm presentedin [24] in terms of the CPU time and used memory, and several other criteria.For all considered examples, the
Maple implementation of the new algorithmsare shown to be superior over the existing ones. One interesting research direc-tion might be to develop a new version of the proposed signature-based versionof the involutive algorithm by incorporating the advantages of the algorithm in[16], in particular of the Janet trees [15]. Furthermore, it would be of interest tostudy the behavior of different possible techniques to improve the computationof Pommaret bases.
Acknowledgments.
The research of the second author was in part supported by a grant from IPM (No. 94550420).
References
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