A Signature-based Algorithm for computing Computing Gröbner Bases over Principal Ideal Domains
AA Signature-based Algorithm for ComputingGröbner Bases over Principal Ideal Domains
Maria Francis and Thibaut Verron
Abstract.
Signature-based algorithms have become a standard approach for Gröbner basis com-putations for polynomial systems over fields, but how to extend these techniques to coefficients ingeneral rings is not yet as well understood.In this paper, we present a proof-of-concept signature-based algorithm for computing Gröb-ner bases over commutative integral domains. It is adapted from a general version of Möller’s al-gorithm (1988) which considers reductions by multiple polynomials at each step. This algorithmperforms reductions with non-decreasing signatures, and in particular, signature drops do not oc-cur. When the coefficients are from a principal ideal domain ( e.g. the ring of integers or the ringof univariate polynomials over a field), we prove correctness and termination of the algorithm, andwe show how to use signature properties to implement classic signature-based criteria to eliminatesome redundant reductions. In particular, if the input is a regular sequence, the algorithm operateswithout any reduction to 0.We have written a toy implementation of the algorithm in Magma. Early experimental resultssuggest that the algorithm might even be correct and terminate in a more general setting, for poly-nomials over a unique factorization domain ( e.g. the ring of multivariate polynomials over a fieldor a PID).
1. Introduction
The theory of Gröbner bases was introduced by Buchberger in 1965 [5] and has since become afundamental algorithmic tool in computer algebra. Over the past decades, many algorithms havebeen developed to compute Gröbner bases more and more efficiently. The latest iteration of suchalgorithms is the class of signature-based algorithms, which introduce the notion of signatures anduse it to detect and prevent unnecessary or redundant reductions. Following early work in [20], thetechnique of signatures was first formally introduced for Algorithm F5 [11], allowing to compute aGröbner basis for a regular sequence without any reduction to zero. Since then, there have been manyresearch works in this direction [13, 2, 7, 8].All these algorithms are for ideals in polynomial rings over fields. Gröbner bases can be definedand computed over commutative rings [1, Ch. 4]. This can be used in many applications, e.g. forpolynomials over ℤ in lattice-based cryptography [12] or for polynomials over a polynomial ring asan elimination tool [21]. Many other examples are described in [18].If the coefficient ring is not a field, there are two ways to define Gröbner bases, namely weakand strong bases. Strong Gröbner bases ensure that normal forms can be computed as in the case offields. But a strong Gröbner basis is in general larger than a weak one, and if the base ring is not aPrincipal Ideal Domain (PID), then some ideals exist which do not admit a strong Gröbner basis. Onthe other hand, weak Gröbner bases, or simply Gröbner bases, always exist for polynomial ideals overa Noetherian commutative ring. They do not necessarily define a unique normal form, but they canbe used to decide ideal membership. If necessary, over a PID, a post-processing phase performingcoefficient reductions can be used to obtain a strong Gröbner basis. This work was started when the first author was supported by the Austrian FWF grant Y464. The second author is supportedby the Austrian FWF grant F5004. a r X i v : . [ c s . S C ] M a y Maria Francis and Thibaut VerronRecent works have focused on generalizing signature-based techniques to Gröbner basis algo-rithms over rings. First steps in this direction, adding signatures to a modified version of Buchberger’salgorithm for strong Gröbner bases over Euclidean rings [17], were presented in [9]. That paper provesthat a signature-based Buchberger’s algorithm for strong Gröbner bases cannot ensure correctness ofthe result after encountering a “signature-drop”, but can nonetheless be used as a prereduction stepin order to significantly speed up the computations.In this paper, we prove that it is possible to compute a weak signature-Gröbner bases of poly-nomial ideals over PIDs (including Euclidean rings) using signature-based techniques. The proof-of-concept algorithm that we present is adapted from the weak Gröbner basis algorithm due toMöller [19] [1, Sec. 4.2], which is designed to compute a basis for a polynomial ideal over any ring,and does so by considering combinations and reductions by multiple polynomials at once. The maindifference with the results of [9] is that we use a stricter definition of regular reductions, effectivelypreventing more reductions from happening, and at the same time adding more polynomials to thebasis.This ensures that no reductions leading to signature-drops can happen in the algorithm, andas a consequence, we prove that the algorithm terminates and computes a signature Gröbner basiswith elements ordered with non-decreasing signatures. This property allows us to examine classicsignature-based criteria, such as the syzygy criterion, the F5 criterion and the singular criterion, andshow how they can be adapted to the case of PIDs. In particular, when the input forms a regularsequence, the algorithm performs no reductions to zero. To the best of our knowledge, this is the firstalgorithm that, given a regular sequence of polynomials with coefficients in a PID, can compute aGröbner basis of the corresponding ideal without any reduction to zero.Möller also presented an efficient algorithm that computes (strong) Gröbner basis for polyno-mial ideals where the coefficients are from Principal Ideal Rings [19, Section 4]. That algorithm skipsthe combinatorial bottleneck of computing saturated sets. Instead, it uses two polynomials to buildS-polynomials and makes use of Gebauer-Möller criteria [15], previously introduced for fields, todiscard redundant S-polynomials.Whenever necessary, for clarity, we shall refer to that algorithm as Möller’s strong algorithm.The algorithm at the center of our focus, computing weak Gröbner bases, will be referred to asMöller’s weak algorithm, or simply Möller’s algorithm.We have written a toy implementation of the algorithms presented, with the F5 and the singularcriteria, in the Magma Computational Algebra System [4], and compared its efficiency, in terms ofnumber of excluded pairs, with Möller’s strong algorithm. Experimentally, on all considered exam-ples, Möller’s (weak) algorithm with signatures does compute and reduce fewer S-polynomials thanMöller’s strong algorithm.Möller’s (weak) algorithm, without signatures, works for polynomial systems over any Noether-ian commutative ring. The signature-based algorithm is only proved to be correct and to terminatefor PIDs, but with very few changes, it can be made to accommodate inputs with coefficients in amore general ring. Interestingly, early experimental data with coefficients in a multivariate polyno-mial ring (a Unique Factorization Domain but not a PID) suggest that the signature-based algorithmmight work over more general rings than just PIDs. For that reason, and because it does not over-complicate the exposition, we choose to present Möller’s algorithms, with and without signatures, intheir most general form, accepting input over any Noetherian commutative ring.
Previous works.
Signature-based Gröbner basis algorithms over fields have been extensively stud-ied, and an excellent survey of those works can be found in [6]. The technical details of most proofscan be found in [22, 10]. The theory of Gröbner bases for polynomials over Noetherian commutativerings dates back to the 1970s [23, 19] and a good exposition of these approaches can be found in[1]. Algorithms exist for both flavors of Gröbner bases: Buchberger’s algorithm [5] computes weakGröbner bases over a PID, and Möller’s weak algorithm [19] extends this approach to Noetheriancommutative rings. As for strong Gröbner bases, they can be computed using an adapted version ofBuchberger’s algorithm [16] or Möller’s strong algorithm [19]. Algorithms for computing signatureGröbner bases over Euclidean rings have been investigated in [9]. Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 3
2. Notations
Let 𝑅 be a Noetherian integral domain, which is assumed to have a unit and be commutative. Let 𝐴 = 𝑅 [ 𝑥 , … , 𝑥 𝑛 ] be the polynomial ring in 𝑛 indeterminates 𝑥 , … , 𝑥 𝑛 over 𝑅 . A monomial in 𝐴 is 𝑥 𝑎 = 𝑥 𝑎 … 𝑥 𝑎 𝑛 𝑛 where 𝑎 = ( 𝑎 , … , 𝑎 𝑛 ) ∈ ℕ 𝑛 . A term is 𝑘𝑥 𝑎 , where 𝑘 ∈ 𝑅 and 𝑘 ≠ . We willdenote all the terms in 𝐴 by Ter( 𝐴 ) and all the monomials in 𝐴 by Mon( 𝐴 ) . We use the notation 𝔞 for polynomial ideals in 𝐴 = 𝑅 [ 𝑥 , … , 𝑥 𝑛 ] and 𝐼 for ideals in the coefficient ring 𝑅 .The notion of monomial order can be directly extended from 𝕂 [ 𝑥 , … , 𝑥 𝑛 ] to 𝐴 . In the rest ofthe paper, we assume that 𝐴 is endowed with an implicit monomial order ≺ , and we define as usualthe leading monomial LM , the leading term LT and the leading coefficient LC of a given polynomial.Given a tuple of polynomials ( 𝑔 , … , 𝑔 𝑠 ) and 𝑖 ∈ {1 , … , 𝑠 } , we will frequently denote, forbrevity, 𝑀 ( 𝑖 ) = LM( 𝑔 𝑖 ) , 𝐶 ( 𝑖 ) = LC( 𝑔 𝑖 ) and 𝑇 ( 𝑖 ) = LT( 𝑔 𝑖 ) = 𝐶 ( 𝑖 ) 𝑀 ( 𝑖 ) .
3. Gröbner Bases in Polynomial Rings over 𝑅 For more details about the contents of this section, one can refer to [1, Chapter 4]. 𝑅 We assume that our coefficient ring 𝑅 is effective in the following sense.(1) There are algorithms for arithmetic operations ( + , ∗ , zero test) in 𝑅 .(2) There is an algorithm LinDecomp : ∙ Input: { 𝑘 , … , 𝑘 𝑠 } ⊂ 𝑅 , 𝑘 ∈ 𝑅 ∙ Output:
TRUE iff 𝑘 ∈ ⟨ 𝑘 , … , 𝑘 𝑠 ⟩ and if yes, 𝑙 , … , 𝑙 𝑠 ∈ 𝑅 such that 𝑘 = 𝑘 𝑙 + ⋯ + 𝑘 𝑠 𝑙 𝑠 .(3) There is an algorithm SatIdeal : ∙ Input: { 𝑘 , … , 𝑘 𝑠 } ⊂ 𝑅 , 𝑘 ∈ 𝑅 ∙ Output: { 𝑙 , … , 𝑙 𝑟 } ⊂ 𝑅 generators of the saturated ideal ⟨ 𝑘 , … , 𝑘 𝑠 ⟩ ∶ ⟨ 𝑘 ⟩ .The condition that an algorithm LinDecomp exists is called linear equations being solvable in 𝑅 in [1,Def. 4.1.5]. Example.
Euclidean rings are effective, because one can implement those algorithms using GCDcomputations and Euclidean reductions. For example over ℤ , LinDecomp ({4} , is ( TRUE , {3}) ,since 12 is in the ideal ⟨ ⟩ and
12 = 3 ⋅ . The output of SatIdeal ({4} , is {2} since ⟨ ⟩ ∶ ⟨ ⟩ = ( ⟨ ⟩ ∩ ⟨ ⟩ ) = ⟨ ⟩ = ⟨ ⟩ .The ring of multivariate polynomials over a field is also effective, using Gröbner bases andnormal forms to perform the same ideal computations. For reduction in fields it is enough to check if the leading term of 𝑓 is divisible by the leadingmonomial of 𝑔 even though the actual reduction happens with the leading term of 𝑔 . Clearly, inrings this is not a sufficient condition : LC( 𝑔 ) may not divide LC( 𝑓 ) even if LM( 𝑔 ) divides LM( 𝑓 ) .Requiring that LT( 𝑔 ) divide LT( 𝑓 ) leads to the notion of strong Gröbner basis, more details can befound in [1, Sec. 4.5].Here we are interested in computing weak Gröbner bases, and we recall the main definitionsin this section. First, following [19, 1], we expand the definition of reduction to allow for a linearcombination of reducers. We define saturated sets [1, Def.4.2.4] (called maximal sets in [19]). Definition 3.1.
Given a tuple of monomials ( 𝑥 𝑎 , … , 𝑥 𝑎 𝑠 ) , the saturated set for a monomial 𝑥 𝑏 w.r.t. ( 𝑥 𝑎 , … , 𝑥 𝑎 𝑠 ) is defined as Sat( 𝑥 𝑏 ; 𝑥 𝑎 , … , 𝑥 𝑎 𝑠 ) = { 𝑖 ∈ {1 , … , 𝑠 } ∶ 𝑥 𝑎 𝑖 ∣ 𝑥 𝑏 } . A set
𝐽 ⊆ {1 , … , 𝑠 } is said to be saturated w.r.t. ( 𝑥 𝑎 , … , 𝑥 𝑎 𝑠 ) if 𝐽 = Sat( 𝑀 ( 𝐽 ); 𝑥 𝑎 , … , 𝑥 𝑎 𝑠 ) where 𝑀 ( 𝐽 ) = lcm( 𝑥 𝑎 𝑖 ∶ 𝑖 ∈ 𝐽 ) . When clear from the context, we shall omit the list of monomials andwrite 𝐽 𝑥 𝑏 = Sat( 𝑥 𝑏 ) .Given a tuple of polynomials ( 𝑓 , … , 𝑓 𝑠 ) and a set of indices 𝐽 ⊂ {1 , … , 𝑠 } , we denote by 𝐼 𝐽 the ideal of 𝑅 defined as 𝐼 𝐽 ∶= ⟨ LC( 𝑓 𝑖 ) ∶ 𝑖 ∈ 𝐽 ⟩ and we define 𝑀 ( 𝐽 ) = lcm(LM( 𝑓 ) , … , LM( 𝑓 𝑠 )) . Maria Francis and Thibaut Verron Definition 3.2.
Let 𝑓 ∈ 𝐴 . Let 𝑓 , … , 𝑓 𝑠 ∈ 𝐴 and 𝑥 𝑎 , … , 𝑥 𝑎 𝑠 ∈ Mon( 𝐴 ) be such that 𝑥 𝑎 𝑖 LM( 𝑓 𝑖 ) =LM( 𝑓 ) for all 𝑖 . We say that we can weakly top reduce 𝑓 by 𝑓 , … , 𝑓 𝑠 ∈ 𝐴 if there exist 𝑙 , … , 𝑙 𝑠 in 𝑅 such that LT( 𝑓 ) = 𝑠 ∑ 𝑖 =1 𝑙 𝑖 𝑥 𝑎 𝑖 LT( 𝑓 𝑖 ) . In our setting we will only perform top reductions, so we will simply call them weak reductions .The outcome of the total reduction step is 𝑔 = 𝑓 − ∑ 𝑠𝑖 =1 𝑙 𝑖 𝑥 𝑎 𝑖 𝑓 𝑖 and the 𝑓 𝑖 ’s are called the weakreducers . A polynomial 𝑓 ∈ 𝐴 is weakly reducible if it can be weakly reduced, otherwise it is weaklyreduced .If 𝑔 is the outcome of reducing 𝑓 , then LM( 𝑔 ) ≺ LM( 𝑓 ) . Example.
Consider the polynomial ring ℤ [ 𝑥, 𝑦 ] with the lex ordering 𝑦 ≺ 𝑥 , and consider the set 𝐹 = { 𝑓 , 𝑓 , 𝑓 , 𝑓 , 𝑓 } in ℤ [ 𝑥, 𝑦 ] , with 𝑓 = 4 𝑥𝑦 + 𝑥, 𝑓 = 3 𝑥 + 𝑦, 𝑓 = 5 𝑥, 𝑓 = 4 𝑦 + 𝑦, 𝑓 = 5 𝑦 .Let 𝑓 = 2 𝑥𝑦 + 13 𝑦 − 5 . We have LT( 𝑓 ) = 2 𝑥𝑦 = (2 𝑦 )LT( 𝑓 ) − (2)LT( 𝑓 ) . This implies we canweakly reduce 𝑓 with 𝑓 , 𝑓 to get 𝑔 = 𝑓 − (2 𝑦𝑓 − 2 𝑓 ) = 2 𝑥 + 13 𝑦 − 5 .We are now prepared to give the definition of (weak) Gröbner bases for an ideal in 𝐴 . Definition 3.3.
Let 𝔞 be an ideal in 𝐴 and 𝐺 = { 𝑔 , … , 𝑔 𝑡 } be a finite set of nonzero polynomials in 𝔞 . The set 𝐺 is called a weak Gröbner basis of 𝔞 in 𝐴 if it satisfies the following equivalent properties.1. ⟨ LT( 𝐺 ) ⟩ = ⟨ LT( 𝔞 ) ⟩ ;2. for any 𝑓 ∈ 𝔞 , 𝑓 is weakly reducible modulo 𝐺 ;3. for any 𝑓 ∈ 𝐴 , 𝑓 ∈ 𝔞 if and only if 𝑓 weakly reduces to modulo 𝐺 . Remark . Even though the notion of weak Gröbner bases is a weaker notion than that of strongGröbner bases, one can use weak polynomial reductions to test for ideal membership. One can alsodefine normal forms modulo a polynomial ideal. However, for those normal forms to be unique, oneneeds to perform further reductions on the coefficients, to “coset representative form”, and one needsto perform reductions on non-leading coefficients as well [1, Th. 4.3.3]. Finally, note that, over a PID,one can easily recover a strong basis from a weak one [19, Th. 4].
In this section, we present Möller’s (weak) algorithm [19] for computing Gröbner bases over ringssatisfying the conditions of Sec. 3.1. This algorithm is analogous to Buchberger’s algorithm for rings,where the polynomial reduction is as defined above and S-polynomials are replaced with linear com-binations of several (possibly more than ) polynomials, defined in the following sense.Consider a set { 𝑔 , … , 𝑔 𝑡 } of polynomials. For 𝑖 ∈ {1 , … , 𝑡 } , let 𝑀 ( 𝑖 ) = LM( 𝑔 𝑖 ) , 𝐶 ( 𝑖 ) =LC( 𝑔 𝑖 ) and 𝑇 ( 𝑖 ) = LT( 𝑔 𝑖 ) . Let 𝐽 be a saturated subset of {1 , … , 𝑡 } w.r.t. { 𝑀 (1) , … , 𝑀 ( 𝑡 )} . Recallthat 𝑀 ( 𝐽 ) = lcm( 𝑀 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 ) . By definition, for all 𝑗 ∈ 𝐽 , 𝑀 ( 𝑗 ) divides 𝑀 ( 𝐽 ) and 𝐽 is maximalwith this property.Let 𝑠 ∈ 𝐽 and 𝐽 ∗ = 𝐽 ⧵ { 𝑠 } . Similar to the idea behind S-polynomials, we want to eliminatethe leading term 𝐶 ( 𝑠 ) 𝑀 ( 𝐽 ) of 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝑔 𝑠 . This can only be done if we multiply 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝑔 𝑠 by an elementof the saturated ideal ⟨ 𝐶 ( 𝑖 ) ∶ 𝑖 ∈ 𝐽 , 𝑖 ≠ 𝑠 ⟩ ∶ ⟨ 𝐶 ( 𝑠 ) ⟩ . We want to consider all such multipliers, so weneed to consider generators of this saturated ideal.Let 𝑐 be such a generator, by definition 𝑐𝐶 ( 𝑠 ) ∈ ⟨ 𝐶 ( 𝑖 ) ∶ 𝑖 ∈ 𝐽 , 𝑖 ≠ 𝑠 ⟩ so there exists ( 𝑏 𝑖 ) 𝑖 ∈ 𝐽 ∗ ∈ 𝑅 such that 𝑐𝐶 ( 𝑠 ) = ∑ 𝑖 ∈ 𝐽 ∗ 𝑏 𝑖 𝐶 ( 𝑖 ) . The (weak) S-polynomial associated with 𝐽 , 𝑠 and 𝑐 , for somesuitable ( 𝑏 𝑖 ) , is defined asS-Pol (( 𝑔 𝑖 ) 𝑖 ∈ 𝐽 ∗ ; 𝑔 𝑠 ; 𝑐 ) = 𝑐 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝑔 𝑠 − ∑ 𝑖 ∈ 𝐽 ∗ 𝑏 𝑖 𝑀 ( 𝐽 ) 𝑀 ( 𝑖 ) 𝑔 𝑖 . If the ring 𝑅 is a PID, the saturated ideal ⟨ 𝐶 ( 𝑖 ) ∶ 𝑖 ∈ 𝐽 , 𝑖 ≠ 𝑠 ⟩ ∶ ⟨ 𝐶 ( 𝑠 ) ⟩ admits a uniquegenerator 𝑐 and we define 𝐶 ( 𝐽 ; 𝑠 ) = LC( 𝑐𝑔 𝑠 ) = 𝑐𝐶 ( 𝑠 ) = lcm(gcd({ 𝐶 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 ∗ }) , 𝐶 ( 𝑠 )) 𝑇 ( 𝐽 ; 𝑠 ) = LT( 𝑐𝑔 𝑠 ) = 𝐶 ( 𝐽 ; 𝑠 ) 𝑀 ( 𝐽 ) . Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 5Then the S-polynomial associated with 𝐽 , 𝑠 , 𝑐 , for some suitable ( 𝑏 𝑖 ) , can be written in the followingform S-Pol (( 𝑔 𝑖 ) 𝑖 ∈ 𝐽 ∗ ; 𝑔 𝑠 ) = 𝑇 ( 𝐽 ; 𝑠 ) 𝑇 ( 𝑠 ) 𝑔 𝑠 − ∑ 𝑖 ∈ 𝐽 ∗ 𝑏 𝑖 𝑀 ( 𝐽 ) 𝑀 ( 𝑖 ) 𝑔 𝑖 . Using this definition of S-polynomials, we recall Möller’s algorithm (Algo. 1) for computinga Gröbner basis of an ideal given by a set of generators over 𝑅 . The correctness and termination ofthis algorithm are shown in [1, Th. 4.2.8 and Th. 4.2.9]. Algorithm 1
Möller’s algorithm [1, Algo. 4.2.2], [19]
Input 𝐹 = { 𝑓 , … , 𝑓 𝑚 } ⊆ 𝐴 ⧵ {0} , ≺ a monomial order on 𝐴 Output 𝐺 = { 𝑔 , … , 𝑔 𝑡 } , a Gröbner basis of ⟨ 𝐹 ⟩ 𝐺 ← 𝐹 , 𝜎 ← , 𝑠 ← 𝑚 while 𝜎 ≠ 𝑠 do ← { subsets of {1 , … , 𝜎 } saturated w.r.t. LM( 𝑔 ) , … , LM( 𝑔 𝜎 ) which contain 𝜎 } for each 𝐽 ∈ do 𝑀 ( 𝐽 ) ← lcm(LM( 𝑔 𝑗 ) ∶ 𝑗 ∈ 𝐽 ) 𝐽 ∗ ← 𝐽 ⧵ { 𝜎 }{ 𝑐 , … , 𝑐 𝜇 } ← SatIdeal ({LC( 𝑔 𝑗 ) ∶ 𝑗 ∈ 𝐽 ∗ } , LC( 𝑔 𝜎 )) // ⟨ 𝑐 , … , 𝑐 𝜇 ⟩ = ⟨ 𝐶 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 ∗ ⟩ ∶ ⟨ 𝐶 ( 𝜎 ) ⟩ for 𝑖 ∈ {1 , … , 𝜇 } do // For PIDs, 𝜇 = 1 𝑝 ← S-Pol (( 𝑔 𝑗 ) 𝑗 ∈ 𝐽 ∗ ; 𝑔 𝜎 ; 𝑐 𝑖 ) 𝑟 ← Reduce ( 𝑝, 𝐺 ) if 𝑟 ≠ then 𝑔 𝑠 +1 ← 𝑟 , 𝐺 ← 𝐺 ∪ { 𝑔 𝑠 +1 } , 𝑠 ← 𝑠 + 1 𝜎 ← 𝜎 + 1 return 𝐺 Algorithm 2
Reduce (Def. 3.2)
Input 𝐺 = { 𝑔 , … , 𝑔 𝑠 } ⊆ 𝐴 ⧵ {0} , ≺ a monomial order on 𝐴 Output 𝑟 result of reducing 𝑝 modulo 𝐺 reducible ← TRUE , 𝑟 ← 𝑝 while reducible is TRUE do 𝐽 ← { 𝑗 ∈ {1 , … , 𝑠 } ∶ LM( 𝑔 𝑗 ) ∣ LM( 𝑟 )} reducible , ( 𝑘 𝑗 ) 𝑗 ∈ 𝐽 ← LinDecomp ({LC( 𝑔 𝑗 ) ∶ 𝑗 ∈ 𝐽 } , LC( 𝑟 )) // If reducible is TRUE , then
LC( 𝑟 ) = ∑ 𝑗 ∈ 𝐽 𝑘 𝑗 LC( 𝑔 𝑗 ) if reducible then 𝑟 ← 𝑟 − ∑ 𝑗 ∈ 𝐽 𝑘 𝑗 LM( 𝑟 )LM( 𝑔 𝑗 𝑔 𝑗 return 𝑟
4. Signatures in 𝐴 𝑚 We consider the free 𝐴 -module 𝐴 𝑚 with basis 𝐞 , … , 𝐞 𝑚 . A term (resp. monomial) in 𝐴 𝑚 is 𝑘𝑥 𝑎 𝐞 𝑖 (resp. 𝑥 𝑎 𝐞 𝑖 ) for some 𝑘 ∈ 𝑅 ⧵ {0} , 𝑥 𝑎 ∈ Mon( 𝐴 ) , 𝑖 ∈ {1 , … , 𝑚 } . In this paper, terms in 𝐴 𝑚 areordered using the Position Over Term (POT) order, defined by 𝑘𝑥 𝑎 𝐞 𝑖 ≺ 𝑙𝑥 𝑏 𝐞 𝑗 ⟺ 𝑖 ⪇ 𝑗 ( or 𝑖 = 𝑗 and 𝑥 𝑎 ≺ 𝑥 𝑏 ) . Given two terms 𝑘𝑥 𝑎 𝐞 𝑖 and 𝑙𝑥 𝑏 𝐞 𝑗 in 𝐴 𝑚 , we write 𝑘𝑥 𝑎 𝐞 𝑖 ≃ 𝑙𝑥 𝑏 𝐞 𝑗 if they are incomparable, i.e. if 𝑎 = 𝑏 and 𝑖 = 𝑗 . Maria Francis and Thibaut VerronGiven a set of polynomials 𝑓 , … , 𝑓 𝑚 ∈ 𝐴 , elements of 𝐴 𝑚 encode elements of the ideal ⟨ 𝑓 , … , 𝑓 𝑚 ⟩ through the 𝐴 -module homomorphism ̄ ⋅ ∶ 𝐴 𝑚 → 𝐴 , defined by setting 𝐞 𝑖 = 𝑓 𝑖 andextending linearly to 𝐴 𝑚 . In particular, ∑ 𝑚𝑖 =1 𝑝 𝑖 𝐞 𝑖 = ∑ 𝑚𝑖 =1 𝑝 𝑖 𝑓 𝑖 .We recall the concept of signatures in 𝐴 𝑚 . Let 𝐩 = ∑ 𝑚𝑖 =1 𝑝 𝑖 𝐞 𝑖 be a module element. Under thePOT ordering, the signature of 𝐩 is 𝔰 ( 𝐩 ) = LT( 𝑝 𝑖 ) 𝐞 𝑖 where 𝑖 is such that 𝑝 𝑖 +1 = … = 𝑝 𝑚 = 0 and 𝑝 𝑖 ≠ .Signatures are of the form 𝑘𝑥 𝑎 𝐞 𝑖 , where 𝑘 ∈ 𝑅, 𝑥 𝑎 ∈ Mon( 𝐴 ) and 𝐞 𝑖 is a standard basis vector.Note that we have two ways of comparing two similar signatures 𝔰 ( 𝜶 ) = 𝑘𝑥 𝑎 𝐞 𝑖 and 𝔰 ( 𝜷 ) = 𝑙𝑥 𝑏 𝐞 𝑗 . We write 𝔰 ( 𝜶 ) = 𝔰 ( 𝜷 ) if 𝑘 = 𝑙 , 𝑎 = 𝑏 and 𝑖 = 𝑗 , and we write 𝔰 ( 𝜶 ) ≃ 𝔰 ( 𝜷 ) if 𝑎 = 𝑏 and 𝑖 = 𝑗 , 𝑘 and 𝑙 being possibly different. If 𝑅 is a field, one can assume that the coefficient is , and so thisdistinction is not important.Note also that when we order signatures, we only compare the corresponding module monomi-als, and disregard the coefficients. This is a different approach from the one used in [9], where bothsignatures and coefficients are ordered.Given a tuple ( 𝜶 , … , 𝜶 𝑠 ) of module elements in 𝐴 𝑚 and 𝑖, 𝑗 ∈ {1 , … , 𝑠 } , we shall frequentlydenote 𝑆 ( 𝑖 ) = 𝔰 ( 𝜶 𝑖 ) for brevity.In order to keep track of signatures we modify Def. 3.2 to introduce the notion of 𝔰 -reduction. Definition 4.1.
Let 𝐩 ∈ 𝐴 𝑚 . We say that we can signature-reduce (or 𝔰 -reduce ) 𝐩 by 𝜷 , … , 𝜷 𝑠 ∈ 𝐴 𝑚 if we can reduce 𝐩 by 𝜷 , … , 𝜷 𝑠 (in the sense of Def. 3.2) and 𝔰 ( 𝑥 𝑎 𝑖 𝜷 𝑖 ) ⪯ 𝔰 ( 𝐩 ) for all 𝑖 = 1 , … , 𝑠 ,where 𝑥 𝑎 𝑖 = LM( 𝐩 )LM( 𝜷 𝑖 ) . We can define similarly 𝔰 -reduced module elements.If 𝔰 ( 𝑥 𝑎 𝑖 𝜷 𝑖 ) ≃ 𝔰 ( 𝐩 ) for some 𝑖 in the above 𝔰 -reduction, then it is called a singular 𝔰 -reductionstep. Otherwise it is called a regular 𝔰 -reduction step.If 𝔰 ( 𝑥 𝑎 𝑖 𝜷 𝑖 ) ≃ 𝔰 ( 𝐩 ) for exactly one 𝑖 and it is actually an equality 𝔰 ( 𝑙 𝑖 𝑥 𝑎 𝑖 𝜷 𝑖 ) = 𝔰 ( 𝐩 ) , it is calleda 𝔰 -reduction step. Remark . For simplicity, we only carry out weak top reductions, and in particular all 𝔰 -reductionsare weak top 𝔰 -reductions. But performing regular 𝔰 -reduction to eliminate trailing terms does notaffect the correctness of the algorithm.Just like 𝔰 -reduction over fields, one can interpret 𝔰 -reduction as polynomial reduction with anextra condition on the signature of the reducers. The difference with fields is that in 𝑅 [ 𝑥 , … , 𝑥 𝑛 ] polynomial reduction is defined differently from the classic polynomial reduction. Additionally, inthe case of fields, all singular 𝔰 -reductions are -singular.The outcome 𝐪 of 𝔰 -reducing 𝐩 is such that LT( 𝐪 ) ≺ LT( 𝐩 ) and 𝔰 ( 𝐪 ) ⪯ 𝔰 ( 𝐩 ) . If 𝐪 is the result ofa regular 𝔰 -reduction, then 𝔰 ( 𝐪 ) = 𝔰 ( 𝐩 ) . In signature-based algorithms, in order to keep track of thesignatures of the basis elements, we only allow regular 𝔰 -reductions. Later, we will also prove thatelements which are -singular 𝔰 -reducible can be discarded. Remark . In [9, Ex. 2], a signature drop appears when 𝔰 -reducing an element of signature 𝑦 𝐞 with an element of signature 𝑦 𝐞 causing the signature to “drop” to 𝑦 𝐞 . With our definition, since weonly compare the module monomial part of the signatures, this is a (forbidden) singular 𝔰 -reduction. Definition 4.4.
Let 𝔞 = ⟨ 𝑓 , … , 𝑓 𝑚 ⟩ be an ideal in 𝐴 . A finite subset of 𝐴 𝑚 is a (weak) signatureGröbner basis (or 𝔰 -GB for short) of 𝔞 if all 𝐮 ∈ 𝐴 𝑚 𝔰 -reduce to zero mod .Given a signature 𝐓 , we say that is a (partial) signature Gröbner basis up to T if all 𝐮 ∈ 𝐴 𝑚 with signature ≺ 𝐓 𝔰 -reduce to 0 mod .Using this definition, we can give the following characterization of 1-singular reducibility,which allows for an easy algorithmic test. Lemma 4.5 (Characterization of -singular 𝔰 -reducibility). Let = { 𝜶 , … , 𝜶 𝑠 } ⊂ 𝐴 𝑚 and 𝐩 ∈ 𝐴 𝑚 such that is a signature Gröbner basis up to signature 𝔰 ( 𝐩 ) . Then 𝐩 is 1-singular 𝔰 -reducible if and only if there exist 𝑗 ∈ {1 , … , 𝑠 } and 𝑘 ∈ 𝑅 and a monomial 𝑥 𝑎 in 𝐴 such that LM( 𝑥 𝑎 𝜶 𝑗 ) = LM( 𝐩 ) and 𝑘𝑥 𝑎 𝔰 ( 𝜶 𝑗 ) = 𝔰 ( 𝐩 ) . Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 7
Proof. If 𝐩 is 1-singular 𝔰 -reducible, then such 𝑗 , 𝑘 and 𝑥 𝑎 exist by definition. Conversely, given such 𝑗 , 𝑘 and 𝑥 𝑎 , if 𝑘𝑥 𝑎 LT( 𝜶 𝑗 ) = LT( 𝐩 ) , then 𝐩 is 1-singular 𝔰 -reducible. If not, then LM( 𝐩 − 𝑘𝑥 𝑎 𝜶 𝑗 ) =LM( 𝐩 ) . Furthermore, 𝔰 ( 𝐩 − 𝑘𝑥 𝑎 𝜶 𝑗 ) ≺ 𝔰 ( 𝐩 ) , so 𝐩 − 𝑘𝑥 𝑎 𝜶 𝑗 𝔰 -reduces to . In particular, there exist ( 𝜇 𝑖 ) 𝑖 ∈{1 , … ,𝑠 } terms in 𝐴 such that for all 𝑖 with 𝜇 𝑖 ≠ , LM( 𝜇 𝑖 𝜶 𝑖 ) = LM( 𝐩 − 𝑘𝑥 𝑎 𝜶 𝑗 ) , LT( 𝐩 − 𝑘𝑥 𝑎 𝜶 𝑗 ) = ∑ 𝑠𝑖 =1 𝜇 𝑖 LT( 𝜶 𝑖 ) and 𝜇 𝑖 𝔰 ( 𝜶 𝑖 ) ⪯ 𝔰 ( 𝐩 − 𝑘𝑥 𝑎 𝜶 𝑗 ) ≺ 𝔰 ( 𝐩 ) . So putting together the two 𝔰 -reductions, weobtain that LT( 𝐩 ) = 𝑘𝑥 𝑎 LT( 𝜶 𝑗 ) + 𝑠 ∑ 𝑖 =1 𝜇 𝑖 LT( 𝜶 𝑖 ) and this is a 1-singular 𝔰 -reduction of 𝐩 . □ We now define (weak) semi-strong signature Gröbner bases , which form a subclass of weak 𝔰 -Gröbner bases. In the case of rings, it is easier to compute them than to directly compute weak 𝔰 -Gröbner bases. Definition 4.6.
Let 𝔞 = ⟨ 𝑓 , … , 𝑓 𝑚 ⟩ be an ideal in 𝐴 . A finite subset of 𝐴 𝑚 is a semi-strongsignature Gröbner basis (or s-s 𝔰 -GB for short) of 𝔞 if, for all 𝐮 ∈ 𝐴 𝑚 , ∙ either 𝐮 is (weakly) regular 𝔰 -reducible modulo ; ∙ or 𝐮 is 1-singular 𝔰 -reducible modulo ; ∙ or 𝐮 = 0 .Given a signature 𝐓 , semi-strong signature Gröbner bases up to 𝐓 are defined similarly by onlyconsidering module elements with signature ≺ 𝐓 . Lemma 4.7 ( [6, Lem. 4.6] ). Let 𝔞 = ⟨ 𝑓 , … , 𝑓 𝑚 ⟩ be an ideal in 𝐴 and let ⊂ 𝐴 𝑚 . Then1. If is a s-s 𝔰 -GB of 𝔞 , then is a 𝔰 -GB of 𝔞 .2. If is a 𝔰 -GB of 𝔞 , then { 𝜶 ∶ 𝜶 ∈ } is a Gröbner basis of 𝔞 .Proof. The definition of a semi-strong Gröbner basis implies that all 𝐮 ∈ 𝐴 𝑚 with 𝐮 ≠ are 𝔰 -reducible modulo , and so such 𝔰 -reductions form a chain which can only terminate at .The proof that a signature Gröbner basis is a Gröbner basis is classical [6, Lem. 4.1]. □ In order to compute signature Gröbner bases, similar to the case of fields, we will restrict thecomputations to regular S-polynomials. For this purpose, we first introduce the signature of a set ofindices, and regular sets.
Definition 4.8.
Let = ( 𝜶 , … , 𝜶 𝑡 ) be a tuple of module elements in 𝐴 𝑚 and a set 𝐽 ⊆ {1 , … , 𝑡 } .For 𝑖 ∈ {1 , … , 𝑡 } , let 𝑀 ( 𝑖 ) = LM( 𝜶 𝑖 ) , and 𝑆 ( 𝑖 ) = 𝔰 ( 𝜶 𝑖 ) . The presignature of 𝐽 is defined as 𝑆 𝐽 = max 𝑠 ∈ 𝐽 { 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝑆 ( 𝑠 ) } . We say that 𝐽 is a regular set if there exists exactly one 𝑠 ∈ 𝐽 such that 𝑆 𝐽 ≃ 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝔰 ( 𝜶 𝑠 ) . Theindex 𝑠 is called the signature index of 𝐽 . We say that 𝐽 is a regular saturated set if 𝐽 ⧵ { 𝑠 } containsall 𝑗 such that 𝑀 ( 𝑗 ) ∣ 𝑀 ( 𝐽 ) and 𝑀 ( 𝐽 ) 𝑀 ( 𝑗 ) 𝑆 ( 𝑗 ) ≺ 𝑆 𝐽 .Note that given a regular set 𝐽 , one can always compute a regular saturated set 𝐽 ′ containing 𝐽 , by adding those indices 𝑗 such that 𝑀 ( 𝑗 ) ∣ 𝑀 ( 𝐽 ) and 𝑀 ( 𝐽 ) 𝑀 ( 𝑗 ) 𝑆 ( 𝑗 ) ≺ 𝑆 𝐽 . Definition 4.9.
Let ( 𝜶 , … , 𝜶 𝑡 ) be a tuple of module elements in 𝐴 𝑚 . For 𝑖 ∈ {1 , … , 𝑡 } , let 𝑀 ( 𝑖 ) =LM( 𝜶 𝑖 ) , 𝐶 ( 𝑖 ) = LC( 𝜶 𝑖 ) and 𝑆 ( 𝑖 ) = 𝔰 ( 𝜶 𝑖 ) . Let 𝐽 ⊂ {1 , … , 𝑡 } be a regular saturated set with signatureindex 𝑠 , and let 𝐽 ∗ = 𝐽 ⧵ { 𝑠 } . Let 𝑐 be an element of a family of generators of ⟨ 𝐶 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 ∗ ⟩ ∶ ⟨ 𝐶 ( 𝑠 ) ⟩ . Let ( 𝑏 𝑗 ) 𝑗 ∈ 𝐽 ∗ be a tuple of elements of 𝑅 such that 𝑐𝐶 ( 𝑠 ) = ∑ 𝑗 ∈ 𝐽 ∗ 𝑏 𝑗 𝐶 ( 𝑗 ) . Then the (weak)S-polynomial associated with 𝐽 and 𝑐 is defined asS-Pol (( 𝑔 𝑗 ) 𝑗 ∈ 𝐽 ; 𝑐 ) = 𝑐 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝜶 𝑠 − ∑ 𝑗 ∈ 𝐽 ∗ 𝑏 𝑗 𝑀 ( 𝐽 ) 𝑀 ( 𝑗 ) 𝜶 𝑗 . Maria Francis and Thibaut VerronIts signature is 𝑆 ( 𝐽 ; 𝑐 ) = 𝔰 ( S-Pol (( 𝑔 𝑗 ) 𝑗 ∈ 𝐽 ; 𝑐 )) = 𝑐𝑆 𝐽 = 𝑐 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝑆 ( 𝑠 ) . Remark . When dealing with regular saturated sets, unlike in Sec. 3.2, we do not need to specifywhich 𝑠 ∈ 𝐽 is singled out when computing the S-polynomial: the only possible 𝑠 is the signatureindex of 𝐽 . Remark . If the coefficient ring is a PID, the ideal ⟨ 𝐶 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 ∗ ⟩ ∶ ⟨ 𝐶 ( 𝑠 ) ⟩ is principal, and 𝑐 is uniquely determined up to an invertible factor. As such, it can be omitted, and in that case weshall simply write S-Pol ( 𝐽 ) for the S-polynomial, and 𝑆 ( 𝐽 ) for its signature. The signature can thenbe written as 𝑆 ( 𝐽 ) = 𝐶 ( 𝐽 ) 𝐶 ( 𝑠 ) 𝑆 𝐽 = 𝐶 ( 𝐽 ) 𝐶 ( 𝑠 ) 𝑀 ( 𝐽 ) 𝑀 ( 𝑠 ) 𝑆 ( 𝑠 ) .
5. Adding signatures to Möller’s weak algorithm
Recall that all 𝔰 -reductions are weak top 𝔰 -reductions. In this section, all S-polynomials are weakS-polynomials. Algorithm
SigMöller (Algo. 3) is a signature-based version of Möller’s algorithm which, given anideal 𝔞 in 𝑅 [ 𝑥 , … , 𝑥 𝑛 ] where 𝑅 is a PID, computes a signature Gröbner basis of 𝔞 .The algorithm proceeds by maintaining a list of regular saturated sets and computing weakS-polynomials obtained from these saturated sets. At each step, it selects the next regular saturated set 𝐽 ∈ such that 𝐽 has minimal presignature amongst elements of . This ensures that the algorithmcomputes new elements for the signature Gröbner basis with nondecreasing signatures (Prop. 5.2).The algorithm then regular 𝔰 -reduces these S-polynomials w.r.t. the previous elements, and addsto the basis those which are not equal to 0 and are not 1-singular 𝔰 -reducible. Signature-based Gröb-ner basis algorithms over fields typically discard all new elements which are singular 𝔰 -reducible,but this may be too restrictive for rings. On the other hand, the proof of Lem. 5.4 justifies that 1-singular 𝔰 -reducible module elements can be safely discarded in the computations. The correctnessof the criterion for 1-singular 𝔰 -reducibility (Algo. 4) was justified in Lem. 4.5. The correctness andtermination of Algorithm SigMöller are proved in Th. 5.5 and Th. 5.6 respectively.Due to space constraints, the subroutine
RegularReduce is not explicitly written. It implementsregular 𝔰 -reduction of a module element 𝐩 w.r.t. a set of module elements { 𝜶 , … , 𝜶 𝑠 } . It is a straight-forward transposition of Reduce (Algo. 2), with the additional condition that we only consider asreducers of 𝐫 those 𝜶 𝑗 with LM( 𝜶 𝑗 ) ∣ LM( 𝐫 ) and LM( 𝐫 )LM( 𝜶 𝑗 ) 𝔰 ( 𝜶 𝑗 ) ≺ 𝔰 ( 𝐫 ) . Remark . Note that the algorithms, as presented, perform computations on module elements.However, for practical implementations, this represents a significant overhead. On the other hand,for any module element 𝜶 , we only need its polynomial value 𝜶 and its signature 𝔰 ( 𝜶 ) . Hence thealgorithm only needs to keep track of the signatures of elements, which is made possible by therestriction to regular S-polynomials and regular 𝔰 -reductions. Example.
An example run of Algorithm 3 is provided in Appendix A.
In this section we prove the correctness of the algorithms presented in Sec. 5.1. The first result statesthat Algorithm
SigMöller computes elements of the signature Gröbner basis in nondecreasing orderon their signatures.
Proposition 5.2.
Let ( 𝜶 , … , 𝜶 𝑡 ) be the value of at any point in the course of Algorithm SigMöller .Then 𝔰 ( 𝜶 ) ⪯ 𝔰 ( 𝜶 ) ⪯ ⋯ ⪯ 𝔰 ( 𝜶 𝑡 ) . Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 9
Algorithm 3
Signature-based Möller’s algorithm (
SigMöller ) Input 𝐹 = { 𝑓 , … , 𝑓 𝑚 } ⊆ 𝐴 ⧵ {0} , ≺ a monomial order on 𝐴 Output = { 𝜶 , … , 𝜶 𝑡 } a semi-strong signature-Gröbner basis of ⟨ 𝐹 ⟩ ← ∅ , 𝜎 ← for 𝑖 ∈ {1 , … , 𝑚 } do 𝐞 ′ 𝑖 ← RegularReduce ( 𝐞 𝑖 , ) if 𝐞 ′ 𝑖 ≠ then = ∪ { 𝐞 ′ 𝑖 } , 𝑠 ← | | // 𝜶 𝑠 = 𝐞 ′ 𝑖 ← { Regular saturated sets of {1 , … , 𝑠 } containing 𝑠 } while ≠ ∅ do Pick and remove from a regular saturated set with minimal presignature 𝑆 𝐽 𝑀 ( 𝐽 ) ← lcm(LM( 𝜶 𝑗 ) ∶ 𝑗 ∈ 𝐽 ) 𝜏 ← signature index of 𝐽𝐽 ∗ ← 𝐽 ⧵ { 𝜏 }{ 𝑐 , … , 𝑐 𝜇 } ← SatIdeal ({LC( 𝜶 𝑗 ) ∶ 𝑗 ∈ 𝐽 ∗ } , LC( 𝜶 𝜏 )) for 𝑖 ∈ {1 , … , 𝜇 } do // For PIDs, 𝜇 = 1 𝑝 ← S-Pol (( 𝑔 𝑗 ) 𝑗 ∈ 𝐽 ; 𝑐 𝑖 ) 𝐫 ← RegularReduce ( 𝐩 , ) if 𝐫 ≠ and not ( 𝐫 , ) then 𝜶 𝑠 +1 ← 𝐫 // 𝜶 𝑠 +1 has signature 𝑆 ( 𝐽 ) = 𝑐 𝑖 𝑆 𝐽 ← ∪ { 𝜶 𝑠 +1 } ← ∪ { Regular saturated sets of {1 , … , 𝑠 + 1} containing 𝑠 + 1} 𝑠 ← 𝑠 + 1 return Proof.
Assume that this is not the case, and let 𝑖 be the smallest index such that 𝔰 ( 𝜶 𝑖 ) ≻ 𝔰 ( 𝜶 𝑖 +1 ) .Let 𝐽 𝑖 (resp. 𝐽 𝑖 +1 ) be the saturated set used to compute 𝜶 𝑖 (resp. 𝜶 𝑖 +1 ). Note that 𝔰 ( 𝜶 𝑖 ) ≃ 𝑆 ( 𝐽 𝑖 ) and 𝔰 ( 𝜶 𝑖 +1 ) ≃ 𝑆 ( 𝐽 𝑖 +1 ) .If 𝑖 ∉ 𝐽 𝑖 +1 , then 𝐽 𝑖 +1 was already in the queue when 𝐽 𝑖 was selected, and so, by the selectioncriterion in the algorithm, 𝑆 ( 𝐽 𝑖 ) ⪯ 𝑆 ( 𝐽 𝑖 +1 ) .If 𝑖 ∈ 𝐽 𝑖 +1 , then 𝑆 ( 𝐽 𝑖 +1 ) ⪰ 𝑥 𝐽𝑖 +1 LM( 𝜶 𝑖 ) 𝔰 ( 𝜶 𝑖 ) ⪰ 𝔰 ( 𝜶 𝑖 ) . □ The following useful lemma gives consequences of the fact that two regular 𝔰 -reduced elementsshare the same signature. Lemma 5.3.
Let = ( 𝜶 , … , 𝜶 𝑠 ) be a signature Gröbner basis up to signature 𝐋 . Let 𝐩 , 𝐪 ∈ 𝐴 𝑚 such that 𝔰 ( 𝐩 ) = 𝔰 ( 𝐪 ) = 𝐋 , and 𝐩 and 𝐪 are regular 𝔰 -reduced. Then LM( 𝐩 ) = LM( 𝐪 ) and either LT( 𝐩 ) = LT( 𝐪 ) , or LC( 𝐩 − 𝐪 ) lies in the ideal 𝐶 ∶= ⟨ LC( 𝜶 𝑗 ) ∶ LM( 𝜶 𝑗 ) ∣ 𝑚 and 𝑚 LM( 𝜶 𝑗 ) 𝔰 ( 𝜶 𝑗 ) ≄ 𝔰 ( 𝐩 ) ⟩ . Proof.
Let 𝐫 = 𝐩 − 𝐪 . Since 𝔰 ( 𝐩 ) = 𝔰 ( 𝐪 ) , we have 𝔰 ( 𝐫 ) ≺ 𝔰 ( 𝐩 ) = 𝐋 , and so 𝐫 𝔰 -reduces to modulo . Assume first that LM( 𝐩 ) ≠ LM( 𝐪 ) , then w.l.o.g. we may assume that LM( 𝐩 ) ≻ LM( 𝐪 ) ,so LM( 𝐫 ) = LM( 𝐩 ) . Since 𝐫 is regular 𝔰 -reducible, 𝐩 is 𝔰 -reducible. This is a contradiction with theassumption that 𝐩 is 𝔰 -reduced.So LM( 𝐩 ) = LM( 𝐪 ) =∶ 𝑚 . If LT( 𝐩 ) ≠ LT( 𝐪 ) , 𝐶 is the ideal of leading coefficients of polyno-mials which can eliminate 𝑚 , and since 𝐫 is 𝔰 -reducible, LC( 𝐩 ) − LC( 𝐪 ) ∈ 𝐶 . □ We now prove the correctness of Algorithm
SigMöller . The proof follows the structure of theproof in the case of fields [22], and adapts it to Möller’s algorithm over PIDs. In particular, it takes intoaccount weak 𝔰 -reductions instead of classical 𝔰 -reductions. The algorithm ensures that all regular0 Maria Francis and Thibaut Verron Algorithm 4
Test of 1-singular 𝔰 -reducibility modulo a partial 𝔰 -GB ( ) Input = { 𝜶 , … , 𝜶 𝑠 } ⊂ 𝐴 𝑚 and 𝐩 ∈ 𝐴 𝑚 such that 𝐩 is regular 𝔰 -reduced w.r.t. and is asignature Gröbner basis up to 𝔰 ( 𝐩 ) Output
TRUE iff 𝐩 is 1-singular 𝔰 -reducible modulo 𝐽 ← { 𝑗 ∈ {1 , … , 𝑠 } ∶ LM( 𝜶 𝑗 ) ∣ LM( 𝐩 ) and LM( 𝐩 )LM( 𝜶 𝑗 ) 𝔰 ( 𝜶 𝑗 ) ⪯ 𝔰 ( 𝐩 ) } return ∃ 𝑗 ∈ 𝐽 , ∃ 𝑘 𝑗 ∈ 𝑅, 𝑘 𝑗 LM( 𝐩 )LM( 𝜶 𝑗 ) 𝔰 ( 𝜶 𝑗 ) = 𝔰 ( 𝐩 ) S-polynomials up to a given signature 𝐓 𝔰 -reduce to 0, and proving the correctness of the algorithmrequires proving that this implies that all module elements with signature ≺ 𝐓 𝔰 -reduce to .The key lemma of the proof is the following. Lemma 5.4.
Let = ( 𝜶 , … , 𝜶 𝑠 ) ⊆ 𝐴 𝑚 . Let 𝐮 ∈ 𝐴 𝑚 ⧵ {0} be 𝔰 -reduced such that 𝐮 ≠ . Assumethat is a s-s 𝔰 -GB basis up to signature 𝔰 ( 𝐮 ) . Then there exists an S-polynomial 𝐩 w.r.t. , suchthat:1. the signature of 𝐩 divides the signature of 𝐮 : 𝑘𝑥 𝑎 𝔰 ( 𝐩 ) = 𝔰 ( 𝐮 ) with 𝑘 ∈ 𝑅 and 𝑥 𝑎 ∈ Mon( 𝐴 ) ;2. if 𝐩 ′ is the result of regular 𝔰 -reducing 𝐩 w.r.t. , then 𝑘𝑥 𝑎 𝐩 ′ is regular 𝔰 -reduced.Proof. The proof is in two steps: first, we construct a S-polynomial 𝐩 whose signature divides 𝔰 ( 𝐮 ) ,and then, starting from 𝐩 , we show that there exists an S-polynomial satisfying the conditions of thelemma.In the remainder of the proof, for 𝑖 ∈ {1 , … , 𝑠 } , let 𝑀 ( 𝑖 ) = LM( 𝜶 𝑖 ) , 𝐶 ( 𝑖 ) = LC( 𝜶 𝑖 ) , 𝑇 ( 𝑖 ) =LT( 𝜶 𝑖 ) and 𝑆 ( 𝑖 ) = 𝔰 ( 𝜶 𝑖 ) . Existence of a S-polynomial satisfying 1.
For the first step, let 𝔰 ( 𝐮 ) be 𝑙𝑥 𝑏 𝐞 𝑖 for some 𝑙 ∈ 𝑅 , 𝑥 𝑏 a monomial and 𝐞 𝑖 a basis vector. Let 𝐞 ′ 𝑖 be the result of regular 𝔰 -reducing 𝐞 𝑖 . If 𝐞 ′ 𝑖 = 0 , then 𝐮 regular 𝔰 -reduces to , which is a contradiction since we assumed 𝐮 to be 𝔰 -reduced and 𝐮 ≠ . Let 𝐋 = 𝑙𝑥 𝑏 𝐞 ′ 𝑖 , it has signature 𝑙𝑥 𝑏 𝐞 𝑖 . Then 𝐮 − 𝐋 has a smaller signature than 𝐮 , so it 𝔰 -reduces to zeroand in particular it is 𝔰 -reducible. Also, 𝐋 is 𝔰 -reducible by 𝐞 ′ 𝑖 . Consider the sum ( 𝐮 − 𝐋 ) + 𝐋 = 𝐮 . Itis not 𝔰 -reducible, which implies that LT( 𝐮 − 𝐋 ) = −LT( 𝐋 ) .Let 𝐽 LM( 𝐋 ) be the maximal regular saturated set 𝐽 with 𝑀 ( 𝐽 ) ∣ LM( 𝐋 ) . Since 𝐮 − 𝐋 𝔰 -reducesto zero, there exists ( 𝑚 𝑗 ) 𝑗 ∈ 𝐽 LM( 𝐋 ) monomials in 𝐴 , and ( 𝑘 𝑗 ) 𝑗 ∈ 𝐽 LM( 𝐋 ) coefficients in 𝑅 such that LT( 𝐮 − 𝐋 ) = ∑ 𝑗 ∈ 𝐽 LM( 𝐋 ) 𝑘 𝑗 𝑚 𝑗 𝑇 ( 𝑗 ) (5.1)with 𝑚 𝑗 𝑀 ( 𝑗 ) = LM( 𝐮 − 𝐋 ) and 𝔰 ( 𝑘 𝑗 𝑚 𝑗 𝜶 𝑗 ) = 𝑘 𝑗 𝑚 𝑗 𝑆 ( 𝑗 ) ⪯ 𝔰 ( 𝐮 − 𝐋 ) ≺ 𝔰 ( 𝐮 ) for all 𝑖 such that 𝑘 𝑗 ≠ .Let 𝜎 be the index of 𝐞 ′ 𝑖 in , that is 𝜶 𝜎 = 𝐞 ′ 𝑖 . Consider the set 𝐽 ′ = { 𝑗 ∶ 𝑚 𝑗 ≠
0} ∪ { 𝜎 } ⊆ 𝐽 LM( 𝐋 ) , itis regular by construction.Let 𝐽 be a regular saturated set containing 𝐽 ′ . Then, since for all 𝑗 ∈ 𝐽 ′ , 𝑀 ( 𝑗 ) ∣ LM( 𝐋 ) = 𝑥 𝑏 𝑀 ( 𝜎 ) , 𝑀 ( 𝐽 ) = lcm { 𝑀 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 ′ } ∣ 𝑥 𝑏 𝑀 ( 𝜎 ) . Furthermore, looking at the leading coefficientsin Eq. (5.2), we have 𝑙 𝐶 ( 𝜎 ) = − ∑ 𝑗 ∈ 𝐽 ′ 𝑘 𝑗 𝐶 ( 𝑗 ) and so 𝑙 ∈ ⟨ 𝐶 ( 𝑗 ) ∶ 𝑗 ∈ 𝐽 , 𝑗 ≠ 𝜎 ⟩ ∶ ⟨ 𝐶 ( 𝜎 ) ⟩ . Since 𝑅 is a PID, this ideal is principal. Let 𝑏 𝐽 beits generator, then 𝑏 𝐽 ∣ 𝑙 . Let 𝐩 be the S-polynomial corresponding to 𝐽 and 𝑏 𝐽 . It is regular byconstruction since 𝐽 is a regular saturated set, and its signature is 𝔰 ( 𝐩 ) = 𝑏 𝐽 𝑀 ( 𝐽 ) 𝑀 ( 𝜎 ) 𝑆 ( 𝜎 ) = 𝑏 𝐽 𝑀 ( 𝐽 )LM( 𝐞 ′ 𝑖 ) 𝐞 𝑖 .Since 𝑏 𝐽 divides 𝑙 and 𝑀 ( 𝐽 ) divides 𝑥 𝑏 𝑀 ( 𝜎 ) , 𝔰 ( 𝐩 ) divides 𝑙𝑥 𝑏 𝑠 𝐞 ′ 𝑖 = 𝔰 ( 𝐋 ) = 𝔰 ( 𝐮 ) . Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 11 Existence of a S-polynomial satisfying 1. and 2.
Let 𝐩 be an S-polynomial whose signature divides 𝔰 ( 𝐮 ) , and let 𝐩 ′ be the regular 𝔰 -reduced form of 𝐩 . Write 𝔰 ( 𝐮 ) = 𝔰 ( 𝑘𝑥 𝑎 𝐩 ) , where 𝑘 ∈ 𝑅 and 𝑥 𝑎 is amonomial.We can assume that 𝑘𝑥 𝑎 𝐩 ′ is regular 𝔰 -reducible or else we are done. We then construct anS-polynomial 𝐪 such that 𝔰 ( 𝑙𝑥 𝑏 𝐪 ) = 𝔰 ( 𝐮 ) and LM( 𝑘𝑥 𝑎 𝐩 ) ≻ LM( 𝑙𝑥 𝑏 𝐪 ) . If 𝑙𝑥 𝑏 𝐪 ′ , where 𝐪 ′ is obtainedby regular 𝔰 -reducing 𝐪 , is not regular 𝔰 -reducible then we are done. Otherwise we can do the sameprocess again and get a third S-polynomial with the same properties and keep repeating. Since theinitial terms are strictly decreasing and we have a well order there are only finitely many such S-polynomials.First, we show that we can assume that 𝑥 𝑎 ≻ . Indeed, assume that 𝑎 = 0 and 𝑘 𝐩 ′ is regular 𝔰 -reducible. Since 𝑅 is an integral domain, LM( 𝑘 𝐩 ′ ) = LM( 𝐩 ′ ) . Let 𝐽 LM( 𝐩 ′ ) be the maximal regularsaturated set 𝐽 with 𝑀 ( 𝐽 ) ∣ LM( 𝐩 ′ ) . Then 𝑘 LC( 𝐩 ′ ) lies in the ideal ⟨ LC( 𝛼 𝑗 ) ∶ 𝑗 ∈ 𝐽 LM( 𝐩 ′ ) ⟩ . Since 𝑅 is a PID, this ideal is principal, let 𝑏 𝐽 LM( 𝐩 ′) be its generator, then 𝑏 𝐽 LM( 𝐩 ′) ∣ 𝑘 . Let 𝐪 be the S-polynomial corresponding to the regular saturated set 𝐽 LM( 𝐩 ′ ) and the generator 𝑏 𝐽 LM( 𝐩 ′) , its signaturedivides 𝔰 ( 𝐮 ) and is strictly divisible by 𝔰 ( 𝐩 ) . Repeating the process as needed, we obtain a strictlyincreasing sequence of elements dividing the coefficient of 𝔰 ( 𝐮 ) , and since 𝑅 is a PID and in particulara unique-factorization domain, this sequence has to be finite. So we can assume that 𝑥 𝑎 ≻ .We will construct two reductions of LT( 𝑘𝑥 𝑎 𝐩 ′ ) , which taken together will give the S-polynomial 𝐪 . For the first reduction, the module element 𝐩 ′ ∈ 𝐴 𝑚 is regular 𝔰 -reduced modulo the s-s 𝔰 -GB ,and its signature is smaller than 𝔰 ( 𝐮 ) . Furthermore, by assumption 𝑘𝑥 𝑎 𝐩 ′ is not regular 𝔰 -reduced, so 𝐩 ′ cannot be . So, by definition of a s-s 𝔰 -GB, 𝐩 ′ is 1-singular 𝔰 -reducible. So there exists ( 𝑡 (1) 𝑖 ) 𝑖 ∈ 𝐽 terms in 𝐴 , with 𝐽 ⊂ {1 , … , 𝑠 } and for all 𝑖 ∈ 𝐽 , 𝑡 (1) 𝑖 ≠ , and such that LT( 𝐩 ′ ) = ∑ 𝑖 ∈ 𝐽 𝑡 (1) 𝑖 LT( 𝜶 𝑖 ) = ∑ 𝑖 ∈ 𝐽 𝑡 (1) 𝑖 𝑇 ( 𝑖 ) (5.2)with for all 𝑖 ∈ 𝐽 , LM( 𝑡 (1) 𝑖 𝜶 𝑖 ) = LM( 𝑡 (1) 𝑖 ) 𝑀 ( 𝑖 ) = LM( 𝐩 ′ ) . Furthermore, there exists 𝜏 in 𝐽 , 𝑡 (1) 𝜏 𝑆 ( 𝜏 ) = 𝔰 ( 𝐩 ) and for all 𝑖 ∈ 𝐽 ⧵ { 𝜏 } , 𝑡 (1) 𝑖 𝑆 ( 𝑖 ) ≺ 𝔰 ( 𝐩 ) .We now build the second reduction. Since 𝑘𝑥 𝑎 𝐩 ′ is regular 𝔰 -reducible, there exists ( 𝑡 (2) 𝑖 ) 𝑖 ∈ 𝐽 terms in 𝐴 , with 𝐽 ⊂ {1 , … , 𝑠 } and for all 𝑖 ∈ 𝐽 , 𝑡 (2) 𝑖 ≠ , such that LT( 𝑘𝑥 𝑎 𝐩 ′ ) = ∑ 𝑖 ∈ 𝐽 𝑡 (2) 𝑖 LT( 𝜶 𝑖 ) = ∑ 𝑖 ∈ 𝐽 𝑡 (2) 𝑖 𝑇 ( 𝑖 ) , (5.3)and for all 𝑗 ∈ 𝐽 , LM( 𝑡 (2) 𝑗 ) 𝑀 ( 𝑗 ) = LM( 𝑘𝑥 𝑎 𝐩 ′ ) and 𝑡 (2) 𝑗 𝑆 ( 𝑗 ) ≺ 𝔰 ( 𝑘𝑥 𝑎 𝐩 ′ ) .Now let 𝐽 = 𝐽 ∪ 𝐽 , and let 𝑡 (1) 𝑖 = 0 if 𝑖 ∈ 𝐽 ⧵ 𝐽 , 𝑡 (2) 𝑗 = 0 if 𝑗 ∈ 𝐽 ⧵ 𝐽 . Note that 𝜏 ∉ 𝐽 , so 𝑡 (2) 𝜏 = 0 . Combining Eqs. (5.2) and (5.2), we obtain a decomposition of 𝑘𝑥 𝑎 𝑡 𝜏 𝑇 ( 𝜏 ) as 𝑘𝑥 𝑎 𝑡 𝜏 𝑇 ( 𝜏 ) = − ∑ 𝑖 ∈ 𝐽 ⧵ { 𝜏 } 𝑡 𝑖 𝑇 ( 𝑖 ) . where for all 𝑖 ∈ 𝐽 , 𝑡 𝑖 = 𝑘𝑥 𝑎 𝑡 (1) 𝑖 − 𝑡 (2) 𝑖 . Furthermore, for all 𝑖 ∈ 𝐽 ⧵ { 𝜏 } , LM( 𝑡 𝑖 ) 𝑀 ( 𝑖 ) = LM( 𝑥 𝑎 𝐩 ′ ) =LM( 𝑥 𝑎 𝑡 𝜏 ) 𝑀 ( 𝜏 ) and 𝑡 𝑖 𝑆 ( 𝑖 ) ≺ 𝔰 ( 𝐩 ) = 𝑘𝑥 𝑎 𝑡 𝜏 𝑆 ( 𝜏 ) .The same argument as the one used, in the first part of the proof, to construct an S-polynomialbased on Eq. (5.2) yields an S-polynomial 𝐪 such that 𝔰 ( 𝐪 ) divides 𝔰 ( 𝐮 ) , say 𝑙𝑥 𝑏 𝔰 ( 𝐪 ) = 𝔰 ( 𝐮 ) . Fur-thermore, since the leading term is eliminated in the construction of an S-polynomial, LT( 𝑙𝑥 𝑏 𝐪 ) ≺ LT( 𝑘𝑥 𝑎 𝐩 ′ ) , which concludes the proof. □ Theorem 5.5 (Correctness of Algorithm
SigMöller ). Let 𝐓 be a term of 𝐴 𝑚 and let = ( 𝜶 , … , 𝜶 𝑠 ) ⊆𝐴 𝑚 be a finite basis as computed by Algo. 3. Assume that all regular S-polynomials 𝐩 with 𝔰 ( 𝐩 ) ≺ 𝐓 𝔰 -reduce to w.r.t. . Then is a semi-strong signature-Gröbner basis up to signature 𝐓 . Proof.
To get a contradiction assume there exists a 𝐮 ∈ 𝐴 𝑚 with 𝔰 ( 𝐮 ) ≺ 𝐓 such that 𝐮 does not 𝔰 -reduce to zero. Assume w.l.o.g. that 𝔰 ( 𝐮 ) is ≺ -minimal such that 𝐮 does not 𝔰 -reduce to zero andalso that 𝐮 is regular 𝔰 -reduced.By Lem. 5.4 there is an S-polynomial 𝐩 with 𝔰 ( 𝑘𝑥 𝑎 𝐩 ) = 𝔰 ( 𝐮 ) with 𝑘 ∈ 𝑅 , 𝑥 𝑎 ∈ Mon( 𝐴 ) . Also, 𝑘𝑥 𝑎 𝐩 ′ is regular 𝔰 -reduced where 𝐩 ′ is the result of regular 𝔰 -reducing 𝐩 .Let 𝐽 LM( 𝐮 ) be the maximal regular saturated set 𝐽 with 𝑀 ( 𝐽 ) ∣ LM( 𝐮 ) . Since 𝔰 ( 𝑘𝑥 𝑎 𝐩 ) = 𝔰 ( 𝐮 ) and both 𝑘𝑥 𝑎 𝐩 ′ and 𝐮 are regular 𝔰 -reduced, we have by Lem. 5.3 that LM( 𝑘𝑥 𝑎 𝐩 ′ ) = LM( 𝐮 ) , andeither LT( 𝑘𝑥 𝑎 𝐩 ′ ) = LT( 𝐮 ) , or LC( 𝐮 − 𝑘𝑥 𝑎 𝐩 ′ ) ∈ ⟨ LC( 𝜶 𝑗 ) ∶ 𝑗 ∈ 𝐽 LM( 𝐮 ) ⟩ . So in either case, there exists ( 𝑡 𝑖 ) 𝑖 ∈ 𝐽 LM( 𝐮 ) terms in 𝐴 , possibly all zero, such that LT( 𝐮 ) − LT( 𝑘𝑥 𝑎 𝐩 ′ ) = ∑ 𝑖 ∈ 𝐽 LM( 𝐮 ) 𝑡 𝑖 LT( 𝜶 𝑖 ) and 𝑡 𝑖 LM( 𝜶 𝑖 ) = LM( 𝐫 ) = LM( 𝐮 ) for all 𝑖 such that 𝑡 𝑖 ≠ .Since 𝐩 ′ is a regular S-polynomial with 𝔰 ( 𝐩 ′ ) ⪯ 𝔰 ( 𝐮 ) ≺ 𝐓 , 𝐩 ′ is 𝔰 -reducible, and so 𝑘𝑥 𝑎 𝐩 ′ is 𝔰 -reducible. So there exists ( 𝜏 𝑖 ) 𝑖 ∈ 𝐽 LM( 𝐮 ) terms in 𝐴 such that LT( 𝑘𝑥 𝑎 𝐩 ′ ) = ∑ 𝑖 ∈ 𝐽 LM( 𝐮 ) 𝜏 𝑖 LT( 𝜶 𝑖 ) , and 𝜏 𝑖 LM( 𝜶 𝑖 ) = LM( 𝑘𝑥 𝑎 𝐩 ′ ) = LM( 𝐮 ) for all 𝑖 such that 𝜏 𝑖 ≠ . So LT( 𝐮 ) = ( LT( 𝐮 ) − LT( 𝑘𝑥 𝑎 𝐩 ′ ) ) + LT( 𝑘𝑥 𝑎 𝐩 ′ ) = ∑ 𝑖 ∈ 𝐽 LM( 𝐮 ) ( 𝑡 𝑖 + 𝜏 𝑖 )LT( 𝜶 𝑖 ) , and 𝐮 is 𝔰 -reducible which is a contradiction. □ The usual proofs of termination of signature-based Gröbner basis algorithms ( e.g. [22, Th. 11]) relyon the fact that all elements which are singular 𝔰 -reducible are discarded in the computations. Algo-rithm SigMöller only discards those which are 1-singular 𝔰 -reducible. For this reason, we adapt theproof of termination of Algorithm RB [10, Th. 20], which handles singular 𝔰 -reducible elements ina different way. Theorem 5.6.
Algorithm
SigMöller terminates.Proof.
Let = ( 𝜶 , … , 𝜶 𝑡 , … ) be the sequence of basis elements computed by SigMöller . By con-struction, for all 𝑡 ≥ , 𝜶 𝑡 is not 𝔰 -reducible by 𝑡 −1 ∶= { 𝜶 , … , 𝜶 𝑡 −1 } , and all 𝐯 ∈ 𝐴 𝑚 with 𝔰 ( 𝐯 ) ≺ 𝔰 ( 𝜶 𝑡 ) 𝔰 -reduce to zero w.r.t. 𝑡 −1 .For 𝑖 ≥ , let 𝑀 ( 𝑖 ) = LM( 𝜶 𝑖 ) , 𝑇 ( 𝑖 ) = LT( 𝜶 𝑖 ) . We define the sig-lead ratio 𝑟 ( 𝜶 𝑖 ) of 𝜶 𝑖 as 𝔰 ( 𝜶 𝑖 ) 𝑀 ( 𝑖 ) .Those ratios are ordered naturally by 𝑠𝑚 ≺ 𝑠 ′ 𝑚 ′ ⟺ 𝑠𝑚 ′ ≺ 𝑠 ′ 𝑚 .We partition into subsets 𝑟 = { 𝜶 𝑖 ∣ 𝑟 ( 𝜶 𝑖 ) ≃ 𝑟 } , where ≃ denotes equality up to a coefficientin 𝑅 . We prove that only finitely many 𝑟 are non-empty, and that they are all finite, hence is finite.First, we prove that only finitely many 𝑟 are non-empty. We do so by counting minimal basiselements, where 𝜶 𝑖 is minimal if and only if there is no 𝜶 𝑗 ∈ with 𝔰 ( 𝜶 𝑗 ) ∣ 𝔰 ( 𝜶 𝑖 ) and 𝑇 ( 𝑗 ) ∣ 𝑇 ( 𝑖 ) . Anon-minimal module element 𝜶 𝑖 is 𝔰 -reducible by { 𝜶 , … , 𝜶 𝑖 −1 } ([22, Lem. 12]), and since all basiselements are regular 𝔰 -reduced by construction, 𝜶 𝑖 is singular 𝔰 -reducible. In particular, there existsat least one 𝜶 𝑗 , 𝑗 < 𝑖 and a monomial 𝑚 with 𝔰 ( 𝑚 𝜶 𝑗 ) ≃ 𝔰 ( 𝜶 𝑖 ) and 𝑚𝑀 ( 𝑗 ) = 𝑀 ( 𝑖 ) , so 𝜶 𝑖 and 𝜶 𝑗 liein the same subset 𝑟 . Hence there are at most as many non-empty 𝑟 ’s as there are minimal basiselements. This is finitely many because 𝐴 and 𝐴 𝑚 are Noetherian.Then we prove by induction on the finitely many non-empty sets 𝑟 that each 𝑟 is finite. Let 𝑟 be a sig-lead ratio, assume that for all 𝑟 ′ < 𝑟 , 𝑟 ′ is finite. Let 𝜶 𝑡 ∈ 𝑟 . If 𝜶 𝑡 is 𝐞 𝑖 for some 𝑖 , Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 13then it only counts for one. Otherwise, let 𝐽 be the regular saturated set, and 𝐩 the corresponding S-polynomial, that SigMöller regular 𝔰 -reduced to obtain 𝜶 𝑡 . Then 𝐩 = ∑ 𝑗 ∈ 𝐽 𝑏 𝑗 𝑀 ( 𝐽 ) 𝑀 ( 𝑗 ) 𝜶 𝑗 for 𝑏 𝑗 ∈ 𝑅 , andthere exists 𝜏 ∈ 𝐽 such that for all 𝑗 ∈ 𝐽 ⧵ { 𝜏 } , 𝑀 ( 𝐽 ) 𝑀 ( 𝑗 ) 𝔰 ( 𝜶 𝑗 ) ≺ 𝑀 ( 𝐽 ) 𝑀 ( 𝜏 ) 𝔰 ( 𝜶 𝜏 ) . Also 𝑇 ( 𝑡 ) ≺ LT( 𝑀 ( 𝐽 ) 𝑀 ( 𝜏 ) 𝜶 𝜏 ) and 𝔰 ( 𝜶 𝑡 ) = 𝑀 ( 𝐽 ) 𝑀 ( 𝜏 ) 𝔰 ( 𝜶 𝜏 ) . So 𝑟 = 𝔰 ( 𝜶 𝑡 ) 𝑀 ( 𝑡 ) ≻ 𝔰 ( 𝜶 𝜏 ) 𝑀 ( 𝜏 ) ≻ 𝔰 ( 𝜶 𝑗 ) 𝑀 ( 𝑗 ) for 𝑗 ∈ 𝐽 ⧵ { 𝜏 } . Hence all 𝜶 𝑗 , 𝑗 ∈ 𝐽 arein some 𝑟 𝑗 with 𝑟 𝑗 < 𝑟 , so for computing elements of 𝑟 , the algorithm will consider at most asmany saturated subsets as there are subsets of ⋃ 𝑟 ′ <𝑟 𝑟 , which is finite by induction. Furthermore,since 𝑅 is a PID and in particular Noetherian, with each saturated subset 𝐽 , the algorithm only buildsfinitely many S-polynomials (actually, it only builds one). So overall, we find that 𝑟 is finite, whichconcludes the proof by induction. □ It is well known in the case of fields that additional criteria can be implemented to detect that a regularS-pair will lead to an element which 𝔰 -reduces to 0. In this section, we show how we can implementthree such criteria, namely the syzygy criterion, the F5 criterion and the singular criterion. Syzygy criteria rely on the fact that, if the signature of an S-polynomialcan be written as a linear combination of signatures of syzygies, then this S-polynomial would be asyzygy itself. Signatures of syzygies can be identified in two ways: ∙ the Koszul syzygy between basis elements 𝐩 and 𝐪 such that 𝔰 ( 𝐩 ) = 𝑚 𝐩 𝐞 𝑖 , 𝔰 ( 𝐪 ) = 𝑚 𝐪 𝐞 𝑗 , 𝑖 < 𝑗 is 𝐩𝐪 − 𝐪𝐩 , and it has signature LT( 𝐩 ) 𝔰 ( 𝐪 ) ; ∙ if a regular S-polynomial 𝐩 𝔰 -reduces to , then 𝔰 ( 𝐩 ) and its multiples are signatures of syzygies;thus, the algorithm may maintain a set of generators of signatures of syzygies by adding to thisset 𝔰 ( 𝐩 ) for each S-polynomial 𝐩 𝔰 -reducing to 0.For regular sequences, all syzygies are Koszul syzygies. Proposition 5.7 (Syzygy criterion).
Assume that 𝐓 is a signature such that all module elements withsignature less than 𝐓 𝔰 -reduce to . Let 𝐩 ∈ 𝐴 𝑚 be such that there exist syzygies 𝐳 , … , 𝐳 𝑘 and terms 𝑚 , … , 𝑚 𝑘 in 𝐴 with 𝔰 ( 𝐩 ) = ∑ 𝑘𝑖 =1 𝑚 𝑖 𝔰 ( 𝐳 𝑖 ) , and 𝔰 ( 𝐩 ) ⪯ 𝐓 . Then 𝐩 regular 𝔰 -reduces to .Proof. Let 𝐫 = 𝐩 − ∑ 𝑘𝑖 =1 𝑚 𝑖 𝐳 𝑖 , then 𝔰 ( 𝐫 ) ≺ 𝔰 ( 𝐩 ) ⪯ 𝐓 so 𝐫 𝔰 -reduces to . But 𝐫 = 𝐩 − ∑ 𝑘𝑖 =1 𝑚 𝑖 𝐳 𝑖 = 𝐩 ,so 𝐩 also 𝔰 -reduces to 0 with reducers of signature at most 𝔰 ( 𝐫 ) ≺ 𝔰 ( 𝐩 ) . □ Koszul syzygies can be eliminated with the same technique, but it is more efficient to use theF5 criterion [22, Sec. 3.3].
Proposition 5.8 (F5 criterion, [11, 3] ). Let 𝐩 ∈ 𝐴 𝑚 with signature 𝜇 𝐞 𝑖 , and let { 𝜶 , … , 𝜶 𝑡 } be asignature Gröbner basis of ⟨ 𝑓 , … , 𝑓 𝑖 −1 ⟩ . Then 𝐩 is a Koszul syzygy if and only if 𝜇 is 𝔰 -reduciblemodulo { 𝜶 , … , 𝜶 𝑡 } .Proof. By definition, 𝐩 is a Koszul syzygy if and only if 𝑚 ∈ LT( ⟨ 𝑓 , … , 𝑓 𝑖 −1 ⟩ ) , and the conclusionfollows by definition of a weak Gröbner basis. □ The singular criterion states that the algorithm only needs to considerone S-polynomial with a given signature. So when computing a new S-polynomial, if there alreadyexists a 𝔰 -reduced module element with the same signature, we may discard the current S-polynomialwithout performing any 𝔰 -reduction. Proposition 5.9 (Singular criterion).
Let = { 𝜶 , … , 𝜶 𝑠 } be a signature Gröbner basis up tosignature 𝐓 . Let 𝐩 ∈ 𝐴 𝑚 be such that there exists 𝜶 𝑖 ∈ with 𝔰 ( 𝜶 𝑖 ) = 𝔰 ( 𝐩 ) and 𝔰 ( 𝐩 ) = 𝔰 ( 𝐓 ) . Then 𝐩 𝔰 -reduces to 0.Proof. Let 𝐩 ′ be the result of regular 𝔰 -reducing 𝐩 w.r.t. . By construction, the basis element 𝜶 𝑖 isregular 𝔰 -reduced w.r.t. . So by Lem. 5.3, LM( 𝐩 ′ ) = LM( 𝜶 𝑖 ) , and applying Lem. 4.5, with 𝑘 = 1 and 𝑥 𝑎 = 1 , shows that 𝐩 ′ is -singular 𝔰 -reducible. The result of that 𝔰 -reduction has signature ≺ 𝔰 ( 𝐩 ) = 𝐓 , so it 𝔰 -reduces to 0. □ ABLE
1. Computation of a grevlex GB of the Katsura-2 system in ℤ [ 𝑋 , 𝑋 , 𝑋 ] Algorithm Pairs / sat. sets S-polynomials Reductions to 0Möller strong 78 20 7
SigMöller (with criteria) 170 13 0T
ABLE
2. Computation of a grevlex GB of the Katsura-3 system in ℤ [ 𝑋 , 𝑋 , 𝑋 , 𝑋 ] Algorithm Pairs / sat. sets S-polynomials Reductions to 0Möller strong 861 246 159
SigMöller (with criteria) 2227 51 0
6. Experimental results and future work
We have written a toy implementation of Algo.
SigMöller , with the F5 and Singular criteria. Weprovide functions LinDecomp and
SatIdeal for Euclidean rings, fields and multivariate polynomialrings.Since our focus is on the feasibility of signature-compatible computations and not their effi-ciency, we give data about the number of considered S-polynomials, saturated sets and reductionsto 0, when computing Gröbner bases over ℤ for the polynomial systems Katsura-2 (Table 1) andKatsura-3 (Table 2). The statistics are compared with a run of Möller’s strong algorithm [19]. Eventhough the proposed algorithm, adapted from Möller’s weak algorithm, considers more saturated setsthan Möller’s strong algorithm, thanks to the signatures, it ends up computing and reducing signifi-cantly less S-polynomials, and no reductions to zero appear.Running Algo. SigMöller on larger examples would require optimizations, but it appears thatthe most expensive step is the generation of the saturated sets, which takes time exponential in thesize of the current basis. This step may be accelerated in different ways. First, it is known that in thecase of PIDs, the reductions of Möller’s algorithm can be recovered from those of Möller’s strongalgorithm [1, Sec. 4.4], which may allow to run the algorithms considering only pairs instead ofarbitrary tuples of polynomials. Additionally, Gebauer and Möller’s criteria for fields can be usedto make Möller’s strong algorithm over PIDs more efficient [19]. We will investigate whether it ispossible to prove that these algorithms are compatible with signatures in the future. Finally, futureresearch will be focused on further signature-based criteria, such as the cover criterion describedin [14] and the more general rewriting criteria.The algorithm accepts as input polynomials over any ring, provided that the necessary routinesare defined. In particular, our implementation can run the algorithms on polynomials on the base ring 𝕂 [ 𝑦 , … , 𝑦 𝑘 ] . On small examples in this setting, it appears that the algorithm terminates and returnsa correct output. Understanding the behavior of SigMöller over UFDs or even more general rings willalso be the focus of future research.
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An Introduction to Gröbner Bases . American Mathematical Society, 7 1994.[2] A. Arri and J. Perry. The F5 Criterion Revised.
Journal of Symbolic Computation , 46(9):1017–1029, 2011.[3] M. Bardet, J.-C. Faugère, and B. Salvy. On the Complexity of the 𝐹 Gröbner Basis Algorithm.
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Acknowledgments
The authors thank C. Eder and anonymous referees for helpful suggestions, M. Ceria and T. Morafor a fruitful discussion on the syzygy paradigm for Gröbner basis algorithms, and M. Kauers for hisvaluable insights and comments all through the elaboration of this work.
Maria FrancisDept of Computer Science and Engineering, Indian Institute of Technology, Hyderabad, India [email protected]
Thibaut VerronInstitute for Algebra, Johannes Kepler University, Linz, Austria [email protected]
Appendix A. Example run of Algorithm
SigMöller (Algo. 3)
As an illustration, consider the ring 𝐴 = ℤ [ 𝑥, 𝑦 ] with the lexicographical ordering with 𝑥 > 𝑦 , andthe ideal generated by 𝑓 = 3 𝑥𝑦 + 𝑥 + 𝑦 and 𝑓 = 𝑥 .The algorithm maintains a signature Gröbner basis 𝐺 and a queue of saturated pairs . Bothare finite ordered sequences (lists), which we denote with square brackets, e.g. 𝐺 = [ 𝐠 , 𝐠 , … , 𝐠 𝑡 ] .The elements of 𝐺 are pairs ( polynomial , signature ) , for which we use the notations 𝐠 𝑖 = ( 𝑔 𝑖 , 𝔰 ( 𝐠 𝑖 )) .To simplify the notations, given a basis 𝐺 = [ 𝐠 , … , 𝐠 𝑡 ] , 𝑚 a monomial of 𝐴 and 𝑘 ∈ ℕ , we use thenotation 𝐽 ( 𝑘 ) 𝑚 = Sat( 𝑚 ; LM( 𝑔 ) , … , LM( 𝑔 𝑘 )) = { 𝑖 ∶ 𝑖 ∈ {1 ..𝑘 } ∶ LM( 𝑔 𝑖 ) ∣ 𝑚 } for the saturated sets.In a saturated set, we will use the notation ∗ to denote those indices which contribute the max-imal signature. For example, in the saturated set 𝐽 = {1 , , ∗ , , ∗ } , we would have 𝑆 𝐽 ≃ 𝑀 ( 𝐽 )LM( 𝑔 ) 𝔰 ( 𝐠 ) ≃ 𝑀 ( 𝐽 )LM( 𝑔 ) 𝔰 ( 𝐠 ) ⪲ 𝑀 ( 𝐽 )LM( 𝑔 𝑖 ) 𝔰 ( 𝐠 𝑖 ) for 𝑖 ∈ {1 , , . If only one index is marked with a star, the saturated set is regular and that index is the signatureindex.The algorithm starts with an empty basis 𝐺 = [] and an empty queue of regular saturated sets = [] . We first add 𝑓 with signature 𝐞 to 𝐺 , that is, we add the element 𝐠 = (3 𝑥𝑦 + 𝑥 + 𝑦 , 𝐞 ) to 𝐺 . We observe that no saturated sets can be formed, so 𝐺 = [ 𝐠 ] is a weak signature Gröbnerbasis of ⟨ 𝑓 ⟩ .We then introduce 𝑓 , with signature 𝐞 . It cannot be reduced modulo 𝐺 , so we add to the basisthe element 𝐠 = ( 𝑥 , 𝐞 ) .To form regular saturated sets, we consider all possible least common multiples of leadingmonomials of 𝑔 𝑖 involving 𝑔 . Here, the set of leading monomials is { 𝑥𝑦, 𝑥 } , and the only non-trivial LCM that we can form is 𝑥 𝑦 . For each least common multiple 𝑚 , the set 𝐽 = 𝐽 (2) 𝑚 = Sat( 𝑚 ; LM( 𝑔 ) , LM( 𝑔 )) = { 𝑖 ∈ {1 ,
2} ∶ LM( 𝑔 𝑖 ) divides 𝑚 } is a saturated set, with 𝑀 ( 𝐽 ) = 𝑚 . Here, for 𝑚 = 𝑥 𝑦 , we get 𝐽 = 𝐽 (2) 𝑥 𝑦 = {1 , ∗ } .We have 𝑀 ( 𝐽 ) = 𝑥 𝑦 = 𝑥 LM( 𝑔 ) = 𝑦 LM( 𝑔 ) . We multiply the corresponding signatures andwe compare: here, 𝑥 𝔰 ( 𝐠 ) = 𝑥 𝐞 ⪱ 𝑦 𝔰 ( 𝐠 ) = 𝑦 𝐞 , so the presignature of 𝐽 is 𝑆 𝐽 = 𝑦 𝐞 , and it isregular with signature index .We now compute a S-polynomial associated to 𝐽 , namely ℎ = 3 𝑦𝑔 − 𝑥𝑔 = − 𝑥 − 𝑥𝑦 with signature 𝔰 ( 𝐡 ) = 3 𝑦 𝔰 ( 𝐠 ) = 3 𝑦 𝐞 . Since LM( ℎ ) = −LM( 𝑔 ) and 𝔰 ( 𝐡 ) ⪲ 𝔰 ( 𝐠 ) , ℎ is regular 𝔰 -reducible modulo 𝐺 , and the result is 𝑔 = − 𝑥𝑦 . It still has signature 𝑦 𝐞 , because we onlyperformed a regular 𝔰 -reduction. We add to 𝐺 the element 𝐠 = (− 𝑥𝑦 , 𝑦 𝐞 ) .The next few steps are identical, so we give a fast-forward version:4. Regular saturated set 𝐽 = 𝐽 (3) 𝑥𝑦 = {1 , ∗ } with 𝑀 ( 𝐽 ) = 𝑥𝑦 and 𝑆 𝐽 = 𝑦 𝐞 → 𝐠 = ( 𝑥𝑦 + 𝑦 , 𝑦 𝐞 ) ;5. Regular saturated set 𝐽 = 𝐽 (4) 𝑥𝑦 = {1 , ∗ } with 𝑀 ( 𝐽 ) = 𝑥𝑦 and 𝑆 𝐽 = 𝑦 𝐞 → 𝐠 = (− 𝑥 + 3 𝑦 − 𝑦 , 𝑦 𝐞 ) ;6. Regular saturated set 𝐽 = 𝐽 (5) 𝑥𝑦 = {1 , , ∗ } with 𝑀 ( 𝐽 ) = 𝑥𝑦 and 𝑆 𝐽 = 𝑦 𝐞 → 𝐠 = (3 𝑦 , 𝑦 𝐞 ) ;7. Regular saturated set 𝐽 = 𝐽 (4) 𝑥𝑦 = {1 , , ∗ } with 𝑀 ( 𝐽 ) = 𝑥𝑦 and 𝑆 𝐽 = 𝑦 𝐞 → 𝐠 = ( 𝑦 , 𝑦 𝐞 ) .Both 𝐽 and 𝐽 were added to after Step 4 (construction of 𝐠 ). But at Step 5, since 𝑆 𝐽 ⪱ 𝑆 𝐽 ,we have to consider 𝐽 first, and keep 𝐽 for later. After Step 5, contains both 𝐽 and 𝐽 , whose Signature-based Algorithm for Computing Gröbner Bases over Principal Ideal Domains 17presignatures are incomparable: 𝑆 𝐽 ≃ 𝑆 𝐽 . So we could have considered 𝐽 before 𝐽 , the resultwould still have been correct.After introducing 𝐠 , the basis 𝐺 has elements: 𝐺 = [(3 𝑥𝑦 + … , 𝐞 ) , ( 𝑥 , 𝐞 ) , (− 𝑥𝑦 , 𝑦 𝐞 ) , ( 𝑥𝑦 + … , 𝑦 𝐞 ) , (− 𝑥 + … , 𝑦 𝐞 ) , (3 𝑦 , 𝑦 𝐞 ) , ( 𝑦 , 𝑦 𝐞 )] . The queue is not empty at this point, but before continuing, we need to form regular saturated setsusing the latest addition 𝐠 .We go through the same process as before to form saturated sets:1. List all possible least common multiples of leading monomials involving LM( 𝑔 ) : those are 𝑦 , 𝑥𝑦 , 𝑥 𝑦 .2. For each of them, compute the corresponding saturated set: ∙ 𝑦 gives 𝐽 (7) 𝑦 = {6 ∗ , ∗ } with presignature 𝑦 𝐞 ; ∙ 𝑥𝑦 gives 𝐽 (7) 𝑥𝑦 = {1 , , , , ∗ , ∗ } , with presignature 𝑥𝑦 𝐞 ; ∙ 𝑥 𝑦 gives 𝐽 (7) 𝑥 𝑦 = {1 , , , , , ∗ , ∗ } with presignature 𝑥 𝑦 𝐞 .None of those 3 saturated sets is regular: there is always a signature collision between 𝐠 and 𝐠 . Forexample, with 𝐽 (7) 𝑥 , 𝔰 ( 𝐠 ) ≃ 𝔰 ( 𝐠 ) ≃ 𝑦 𝐞 .So we need to make them regular, which is done by forming new sets with just one of thecolliding signatures. From 𝐽 (7) 𝑦 , we could form {6 ∗ } and {7 ∗ } , which are trivial.From 𝐽 (7) 𝑥𝑦 , we can form the regular saturated sets {1 , , , , ∗ } and {1 , , , , ∗ } . Since theset {1 , , , , ∗ } does not contain , it is already in . And we add the new regular saturated set {1 , , , , ∗ } to .Similarly, from 𝐽 (7) 𝑥 𝑦 , we find the new regular saturated set {1 , , , , , ∗ } to add to .Then we continue with the regular saturated set 𝐽 = {1 , , , ∗ } with 𝑀 ( 𝐽 ) = 𝑥 𝑦 and 𝑆 ( 𝐽 ) = 27 𝑦 𝐞 . It gives rise to ℎ = 3 𝑦 + 𝑦 with signature 𝔰 ( 𝐡 ) = 27 𝑦 𝐞 .Since LM( ℎ ) = 𝑦 LM( 𝑔 ) and 𝔰 ( 𝐡 ) = 𝑦 𝔰 ( 𝐠 ) , we know that ℎ is -singular reducible mod-ulo 𝐺 and can be discarded. Note that we only needed to compare the leading monomials (withoutcoefficients) of ℎ and 𝑔 , and not verify whether there is an actual linear combination eliminatingthat term. Remark
A.1 . If we had considered 𝐽 before 𝐽 at Step 6, 𝐠 would have been built before 𝐠 , and 𝐠 would have been discarded for being 1-singular reducible modulo 𝐠 . In that case, the non-regularsaturated sets 𝐽 (7) 𝑦 , 𝐽 (7) 𝑥𝑦 and 𝐽 (7) 𝑥 𝑦 would never have been considered.The remainder of the run proceeds differently, depending on whether the F5 criterion (Prop. 5.8)is implemented. If it is not, the remaining regular saturated sets all give rise to polynomials regular 𝔰 -reducing to , and the algorithm terminates, returning the -elements basis written above.On the other hand, if the F5 criterion is implemented, those reductions to zero are excluded.Let us illustrate it with the next saturated set in the queue: 𝐽 = {2 , ∗ } , with signature 𝑥𝑦 𝐞 . Weneed to test whether 𝑥𝑦 lies in the ideal of leading terms of ⟨ 𝑓 ⟩ , which is equivalent to testingwhether 𝑥𝑦 is reducible modulo the already computed basis 𝐺 = {(3 𝑥𝑦 + 𝑥 + 𝑦 , 𝐞 )})}