An efficient reduction strategy for signature-based algorithms to compute Groebner basis
aa r X i v : . [ c s . S C ] N ov An efficient reduction strategy for signature-basedalgorithms to compute Gröbner basis
Kosuke Sakata ∗† Graduate School of Environment and Information SciencesYokohama National UniversityYokohama Kanagawa [email protected] 3, 2018
Abstract
This paper introduces a strategy for signature-based algorithms to computeGröbner basis. The signature-based algorithms generate S-pairs instead of S-polynomials,and use s -reduction instead of the usual reduction used in the Buchberger algorithm.There are two strategies for s -reduction: one is the only-top reduction strategy whichis the way that only leading monomials are s -reduced. The other is the full reduc-tion strategy which is the way that all monomials are s -reduced. A new strategy,which we call selective-full strategy, for s -reduction of S-pairs is introduced in thispaper. In the experiment, this strategy is efficient for computing the reduced Gröb-ner basis. For computing a signature Gröbner basis, it is the most efficient or notthe worst of the three strategies. The Gröbner basis algorithm using signature was first introduced by Faugère [1]. The algo-rithm, F5, decreases the number of times of reducing S-pairs (an analogy of S-polynomials)to zero by removing useless critical pairs comparing to existing other algotrithms. Afterthat, several signature-based algorithms to compute Gröbner basis were introduced suchthat F5C [2], GVW [3] and so on. The paper [4] by Eder and Faugère compiled the thesisrelated to signature-based algorithms. We can overview the signature-based algorithmsby the paper.In the signature-based algorithms, for removing useless critical pairs, S-pairs are notreduced like the Buchberger algorithm, but reduced with more restrictions. That re-stricted reduction, called s -reduction, is indispensable in the algorithms. There are two ∗ I would like to thank Professor Shushi Harashita for his helpful comments. † I am grateful to Professor Kazuhiro Yokoyama and Professor Masayuki Noro for discussions on thetopic of this manuscript. s -reducing S-pairs according to the paper [4]. One is the only-top reductionstrategy ( Algorithm 3 ): after generating an S-pair, regular s -reduce leading monomialsuntil the leading monomial cannot be regular s -reduced. The other is the full reductionstrategy ( Algorithm 4 ): after generating an S-pair, regular s -reduce the all monomialsincluded in the S-pairs. It cannot be said that either is efficient because it depends onpolynomial systems we solve and on strategies we use.For efficient Gröbner basis computation, to decrease the number of reductions is one ofthe significant problems, because the reduction process accounts for a large propotion inthe computation. This paper introduces a new strategy for s -reduction aiming to decreasethe number of sum of s -reductions and usual reductions. Overview of the starategy isfollowing: after generating an S-pair, we fulfill only-top reduction. If the S-pair meetsa certain condition( SF in §4), we execute full reduction. We name the strategy the selective-full reduction strategy ( Algorithm 5 ). Efficiency of the strategy was evaluatedby some Gröbner basis benchmarks. The selective-full strategy process fewer times ofreduction for computing the reduced Gröbner basis. For computing a signature Gröbnerbasis, it is the most efficient or not the worst of the three strategies.
In this section, we review the definitions around the signature-based algorithms. Let R bea polynomial ring over a field K , let f , f , . . . , f m ∈ R , let e , e , . . . , e m be the standardbasis of a free module R m . Consider the following homomorphism¯: R m −→ Rα = m X i =1 a i e i α = m X i =1 a i f i , where a , . . . , a m ∈ R , especially e i = f i .We choose a monomial order ≤ on R , and choose a module order (cid:22) which is compatiblewith the monomial order, namely we require that for all monomials a, b ∈ R , a ≤ b if andonly if a e i (cid:22) b e i for i = 1 , . . . , m . An element of R m of the form a e i for a monomial a of R is called a term of R m . The following POT(position over term) order is one of examplesof module orders: let a e i , b e j be two module terms in R m , a e i (cid:22) POT b e j if and only ifeither i < j or i = j and a < b . For α ∈ R m , the signature s ( α ) of α is defined to be theleading term of α with respect to the module order. For f ∈ R ,we denote by LT( f ) of f the leading term with respect to the monomial order. We denote by T( f ) the set of themonomials in f .The S- pair of α, β ∈ R m is defined to bespair( α, β ) = λ LT( α ) α − λ LT( β ) β, where λ is the least common monomial as λ = lcm(LT( α ) , LT( β ))Let G be a subset of R m , let α, α ′ ∈ R m , we say that α is an one-time s -reduced to α ′ if there exist β ∈ G and b ∈ R satisfying: 2 lgorithm 1 TOP_REDUCE input : a finite subset G ∈ R m , an S-pair α output : an S-pair α for β ∈ G doif LT( β ) | LT( α ) and s ( α ) ≻ s ( LT( α )LT( β )) · β ) then α ← α − LT( α )LT( β )) · β return α end ifend forreturn α Algorithm 2
TAIL_REDUCE input : a finite subset G ∈ R m , an S-pair α output : an S-pair α for t ∈ T( α − LT( α )) do ( t is a monomial in α − LT( α )) for β ∈ G doif LT( β ) | t and s ( α ) ≻ s ( t LT( β )) · β ) then α ← α − t LT( β )) · β return α end ifend forend forreturn α (a) LT( bβ ) = t for a (certain) monomial in α (b) s ( bβ ) (cid:22) s ( α ) (c) α ′ = α − bβ .At this time, we call β a reducer. We say that α is s -reduced to α ′′ with respect to G if there exists a sequence α = α (0) , α (1) , · · · , α ( l ) = α ′′ of R m such that α ( i ) is one-time s -reduced to α ( i +1) with respect to G for i = 0 , , · · · , l −
1. A s -reduction is calleda singular s -reduction , if there exists c ∈ K such that s ( bβ ) = c s ( α ) and otherwise itis called a regular s -reduction . If there exists c ∈ K such that LT( bβ ) = c LT( α ), the s -reduction is called a one-time top s -reduction and otherwise called a one-time tail s -reduction . Algorithm 1 shows that if an S-pair α can be one-time top s -reduced by G ,it returns one-time top s -reduced α by G . Algorithm 2 shows that if an S-pair α can beone-time tail s -reduced by G , it returns one-time tail s -reduced α by G . If the α ∈ R m cannot be regular top s -reduced, we say that α is completely regular top s -reduced . Ifall the monomials in α cannot be regular s -reduced, we call α is completely regular full s -reduced . Below, to distinguish from s -reductions, the reductions used in Buchbergeralgorithms are called usual reductions.A subset G ⊆ R m is a signature Gröbner basis with respect to (cid:22) if all α ∈ R m are s -reduced to zero with respect to G . The signature-based algorithms compute a signature3 lgorithm 3 ONLY-TOP_REDUCE input : a finite subset G ∈ R m , an S-pair α output : an S-pair α which is completely regular top s -reduced by Gβ ← while β = α do β ← αα ← TOP_REDUCE(
G, α ) end whilereturn α Algorithm 4
FULL_REDUCE input : a finite subset G ∈ R m , an S-pair α output : an S-pair α which is completely regular full s -reduced by Gβ ← while β = α do β ← αα ← TOP_REDUCE(
G, α ) end while β ← while β = α do β ← αα ← TAIL_REDUCE(
G, α ) end whilereturn α Gröbner basis. If G is a signature Gröbner basis, then { g | g ∈ G } is a Gröbner basisof an ideal of { g | g ∈ G }. The number of the elements of a signature Gröbner basis isalways larger than or equal to that of minimal Gröbner basis. s -reduction strategies In this section, we review the two strategies of s -reducing S-pairs mentioned in the paper[4]. One is the only-top reduction strategy: after generating an S-pair, regular s -reduceleading monomials until the leading monomial cannot be regular s -reduced. The algorithmis represented as Algorithm 3 . By this procedure, the S-pair is completely regular top s -reduced. The other is the full reduction strategy: after generating S-pairs, regular s -reducethe monomials included in the S-pairs. The algorithm is represented as Algorithm 4 .At first, it execute top s -reduction, then if the S-pair is completely regular top s -reduced,execute regular tail s -reduction. By this procedure, the S-pair is completely regular full s -reduced.Each strategy has advantages and disadvantages. When we choose the only-top reduc-tion strategy, it is expected that times of s -reduction is fewer because we regular s -reducethe only top monomials. However, since the terms of the polynomials used to regular s -4 lgorithm 5 SELECTIVE-FULL_REDUCE input : a finite subset G ∈ R m , an S-pair α output : an S-pair α which is completely regular selective-full s -reduced by Gβ ← while β = α do β ← αα ← TOP_REDUCE(
G, α ) end whilefor γ ∈ G doif LT( γ ) | LT( α ) thenreturn α (if α does not satisfy SF , return α ) end ifend for β ← while β = α do β ← αα ← TAIL_REDUCE(
G, α ) end whilereturn α reduce the S-pairs are large with respect to the fixed monomial order, there is a possibilitythat the number of times of s -reduction may increase even if only the regular s -reductionof the leading term is performed. On the other hand, when we choose the full reductionstrategy, we regular s -reduce all terms included in the S-pair, so the terms included inthe S-pairs is relatively small with respect to the fixed monomial order. Also, times ofinterreductions for computing the reduced Gröbner basis becomes few because regular tail s -reductions has executed in advance. However, regular tail s -reductions are restrictedreductions, so it cannot reduce terms sufficiently in comparison with usual reductions.Moreover, number of elements of a signature Gröbner basis tend to be much larger thanthat of minimal Gröbner basis. So, the number of S-pairs that we need to completelyregular full s -reduce is also much larger. s -reduction strategy Consider the case where we compute the reduced Gröbner basis after a signature Gröbnerbasis is computed. The signature-based algorithms compute the signature Gröbner basiswhich is larger than the minimnal Gröbner basis. Therefore, first, compute a minimalGröbner basis from the found signature Gröbner basis. The method is to remove α ∈ G satisfying the following condition from the found signature Gröbner basis: There exists α ′ ∈ G, LT( α ) | LT( α ′ ). Then, we obtain a minimal Gröbner basis. By interreducing thefound minimal Gröbner basis, the reduced Gröbner basis is obtained.Here we consider the relation between the full reduction strategy and reduced Gröbnerbasis. The full reduction strategy can be seen as a strategy to decrease the number oftimes of interreduction. In that sense, there is no need to tail reduction for S-pairs that5ill be removed at the step of computing a minimal Gröbner basis. An algorithm basedon this idea to s -reduce an S-pair is Algorithm 5 . In this algorithm, first, regular top s -reduce the S-pair α until the S-pair becomes completely regular top s -reduced. Then,perform regular tail s -reduction only when the following condition is satisfied:for all α ′ ∈ G, LT( α ′ ) ∤ LT( α ) SF We call this strategy the selective-full reduction . The output of the
Algorithm 5 denotesa completely regular selective-full reduced
S-pair.Following shows that the selective-full strategy is reasonable. (1)
Let α be an S-pair which does not satisfy SF . We can foresee that α will beremoved when we compute a minimal Gröbner basis. If we choose the selective-full strategy, we do not regular tail s -reduce α that is finally discarded. Therefore,it is expected that number of times of s -reductions by the selective-full reductionstrategy is smaller that by full reduction strategy. (2) Consider the case that a signature Gröbner basis has been computed, and then, weobtain an minimal Gröbner basis. If we choose the selective-full strategy, all elementsin the minimal Gröbner basis were completely regular full s -reduced. Therefore, itis expected that number of times of interreductions by the selective-full reductionstrategy is much smaller than that by only-top reduction strategy. (3) Let α ∈ G be a possible reducer for a certain S-pair, and α was not completelyregular full s -reduced, that means that α did not satisfy SF . Then, there exists α ′ such that LT( α ′ ) | LT( α ) and α ′ was completely regular full s -reduced or an inputmodule of the algorithm. Because, in the above situation, there exists α ′ ∈ G suchthat LT( α ′ ) | LT( α ) and LT( β ) ∤ LT( α ′ ) for all β ∈ G . Then, α ′ was generatedfrom a certain S-pair that satisfy SF , or is an input module of the algorithm. So,if the S-pair can be regular s -reduced, a reducer which is regular full s -reduced ispossible to be selected. Therefore, to some extent, number of times of s -reductionis expected to be less number compared to the only-top reduction strategy. In this section, we evaluate the proposed s -reduction strategy, selective-full reductionstrategy, using the well-known benchmark for Gröbner basis. The implementation is doneby C for counting s -reductions, usual reductions and multiplications. The benchmarkwas carried out in each of homogeneous ideals and inhomogeneous ideals. We comparedthree strategies, only-top reduce, full reduce, and selective-full reduce. We refer to [4]for the concept of the signature-based algorithm and the words used below. All systemsare computed over a field of characteristic 32003, with graded reverse lexicographicalmonomial order. For a module order, we used the POT order which is used in the originalF5. For finding the syzygy modules, we used signatures which are zero reduced. Therefore,like F5, all algorithms proceeds incrementally. Like F5C, the reduced Gröbner basis wasfound at each incremental steps. For a rewrite order, we used each ADD and RAT.6 emark. In the