On Two Signature Variants Of Buchberger's Algorithm Over Principal Ideal Domains
aa r X i v : . [ c s . S C ] F e b On Two Signature Variants Of Buchberger’s AlgorithmOver Principal Ideal Domains
Maria Francis
Indian Institute of Technology HyderabadHyderabad, [email protected]
Thibaut Verron
Institute for Algebra / Johannes Kepler UniversityLinz, [email protected]
ABSTRACT
Signature-based algorithms have brought large improvements inthe performances of Gröbner bases algorithms for polynomial sys-tems over fields. Furthermore, they yield additional data which canbe used, for example, to compute the module of syzygies of an idealor to compute coefficients in terms of the input generators.In this paper, we examine two variants of Buchberger’s algo-rithm to compute Gröbner bases over principal ideal domains, withthe addition of signatures. The first one is adapted from Kandri-Rody and Kapur’s algorithm [16], whereas the second one usesthe ideas developed in the algorithms by L. Pan [22] and D. Licht-blau [17]. The differences in constructions between the algorithmsentail differences in the operations which are compatible with thesignatures, and in the criteria which can be used to discard ele-ments.We prove that both algorithms are correct and discuss their rel-ative performances in a prototype implementation in Magma.
CCS CONCEPTS • Computing methodologies → Algebraic algorithms . KEYWORDS
Algorithms, Gröbner bases, Signature-based algorithms, Polynomi-als over rings, Principal Ideal Domains
ACM Reference Format:
Maria Francis and Thibaut Verron. 2021. On Two Signature Variants OfBuchberger’s Algorithm Over Principal Ideal Domains. In . ACM, New York,NY, USA, 9 pages. https://doi.org/XX.XXX/XXXXXX.XXXXXX
Gröbner bases over fields, introduced by Buchberger [4], is a fun-damental tool in computational ideal theory and algebraic geom-etry. Very early on, several approaches were proposed to extendthe algorithmic theory of Gröbner bases to polynomial rings overrings, a summary of which can be found in [1, 2]. Ideals in poly-nomial rings over rings have several applications, for instance in
T. Verron was supported by the Austrian FWF grant P31571-N32.Permission to make digital or hard copies of part or all of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).
Conference’21, July 2021, Washington, DC, USA © 2021 Copyright held by the owner/author(s).ACM ISBN XXXXXXXXXXXXXXXXXXX.https://doi.org/XX.XXX/XXXXXX.XXXXXX number theory [18], or in lattice-based cryptography, where cer-tain residue class polynomial rings over Z called ideal lattices havebeen used [11, 19].There are two ways to define Gröbner bases (GB) over rings,namely weak and strong Gröbner bases, corresponding to two no-tions of reductions. Of the two, strong GB and reductions are themost similar to fields and the ones we consider in this work. It al-lows to efficiently compute the normal form of an element; overprincipal ideal domains (PID’s), all ideals admit a strong GB. Dif-ferent algorithms have been proposed for computing strong basesover PID’s [20, 22] and over Euclidean domains [7, 8, 16, 17].Buchberger’s original algorithm for computing Gröbner basesover fields proceeds by computing and reducing S-polynomials.Over rings, the computation of Gröbner bases additionally requiresto compute so-called G-polynomials, namely combinations of poly-nomials which use Bézout coefficients to make the leading coeffi-cient as small as possible. Kandri-Rody and Kapur’s algorithm [16]was designed for Euclidean domains but works without any mod-ification over PIDs; it proceeds by computing, for each pair of el-ements, both their S- and G-polynomials, and adding them to thequeue for later processing. Pan’s algorithm [22] for PIDs, later re-fined by Lichtblau [17] for Euclidean domains, observes that foreach pair, only one polynomial, S- or G-, is required.Over fields, it was rapidly noticed that many of the reductionsin Buchberger’s algorithm are useless, i.e. , they eventually reach0 and are discarded. Optimizations of Buchberger’s algorithm thatstarted with Buchberger [5] have focused on how to detect theseuseless reductions beforehand [21]. A breakthrough came in theearly 2000s with the class of so-called signature-based algorithmssuch as F5 [10] and later GVW [14]. A comprehensive survey ofsignature-based algorithms can be found in [6]. These algorithmskeep track of, for each computed polynomial, its signature, namelythe leading term of a representation of the polynomial in terms ofthe generators of the ideal. This information can be used to detectreductions to 0, and avoid redundant computations.Furthermore, the computation of a Gröbner basis with signa-tures allows to recover the coefficients of the elements of the basisin terms of the generators, and to compute the module of syzygiesof those generators, without the extra cost of module computationsor additional variables [14].The natural next step is to see whether these signature-basedtechniques can be generalized to Gröbner basis algorithms overrings. In this direction, a hybrid algorithm was presented in [7]that added signatures to a modified version of Kandri-Rody andKapur’s algorithm. The authors showed with a counter-examplethat implementing totally ordered signatures for rings cannot en-sure that the signatures will never decrease/drop during the course onference’21, July 2021, Washington, DC, USA Maria Francis and Thibaut Verron of computing the strong GB, which is the key invariant of mostsignature-based algorithms. The signature-based techniques of [7]could however be used as an efficient preprocessing step to fastenthe computations, falling back to the classical techniques when adrop in signature is detected.In [12], the authors described a theoretical algorithm that com-putes a weak Gröbner basis with signatures, over PID’s, withoutany signature drop, by using a partial order on the signatures. Inthis work, we use similar constructions to adapt Kandri-Rody andKapur’s algorithm and Pan/Lichtblau’s algorithm to the computa-tion of signature Gröbner bases. For that purpose, we need two con-structions: a restriction on the construction of G-polynomials en-suring that we can keep track of their signatures without discard-ing any; and an analogous construction using Bézout coefficientsto obtain elements with small signatures. In the case of Kandri-Rody and Kapur’s algorithm, we prove that the powerful cover cri-terion described in [13] can be applied to eliminate some S-polynomials.In the case of Pan/Lichtblau’s algorithm, the nature of the pairs be-ing computed forces us to relax the restrictions on S-polynomials,and limits the scope of the criteria. In both cases, we prove thatthe algorithms are correct and compute both a signature-Gröbnerbasis of the ideal, and a basis of the signatures of its syzygies.We have implemented both algorithms in the computer algebrasystem Magma [3], with additional optimizations and criteria, andobserve that the relaxed restrictions in Pan/Lichtblau tend to leadto the computation of more pairs. We also compare the time takenfor computing the signature Gröbner basis and using it to recoverinformation on the module, with
Magma implementations of ad-hoc functions for that purpose, and show that using signatures al-lows for a significant speed-up of those operations.
Let N be the set of all non-negative integers. Let 𝑅 be a principalideal domain (PID) that has a unit element and is commutative.We assume that 𝑅 is effective in the sense that one can performall the arithmetic operations in 𝑅 , obtain the gcd of elements andcompute Bézout coefficients. A typical example of such a ring isthe ring of integers Z , with Euclid’s algorithm and the extendedEuclid’s algorithm.Let 𝐴 = 𝑅 [ 𝑥 , . . . , 𝑥 𝑛 vars ] be the polynomial ring in 𝑛 vars inde-terminates 𝑥 , . . . , 𝑥 𝑛 vars over 𝑅 . A monomial in 𝐴 is an elementof the form 𝑥 𝑎 . . . 𝑥 𝑎 𝑛 vars 𝑛 vars where 𝑎 = ( 𝑎 , . . . , 𝑎 𝑛 vars ) ∈ N 𝑛 vars . Aterm in 𝐴 is 𝑎𝜇 , where 𝑎 ∈ 𝑅 \ { } and 𝜇 is a monomial. The set ofterms (resp. monomials) of 𝐴 is denoted by Ter ( 𝐴 ) (resp. Mon ( 𝐴 ) ).A monomial order is an order on Mon ( 𝐴 ) which is compatiblewith multiplication and well-founded. In the rest of the paper, weassume that 𝐴 is endowed with an implicit monomial order < , andwe define as usual the leading monomial lm , the leading term lt and the leading coefficient lc of a given polynomial. By convention,we set lm ( ) = lt ( ) = lc ( ) = .Given a pair of polynomials ( 𝑓 , 𝑔 ) , we denote lcmlm ( 𝑓 , 𝑔 ) (resp.lcmlt ( 𝑓 , 𝑔 ) ) the least common multiple of the leading monomials(resp. leading terms) of 𝑓 and 𝑔 .Given a set of polynomials 𝑓 , . . . , 𝑓 𝑛 polys in 𝐴 , we consider thefree module M = 𝐴 𝑛 polys with basis e , . . . , e 𝑛 polys . For 𝛼 ∈ M with 𝛼 = ( 𝛼 , . . . , 𝛼 𝑛 polys ) , we define 𝛼 = Í 𝛼 𝑖 𝑓 𝑖 . We define the module I = {( 𝛼, 𝛼 ) : 𝛼 ∈ M } ⊂ 𝐴 𝑛 polys + . The module I is isomorphic to M , and in particular it is free withbasis ( e , 𝑓 ) , . . . , ( e 𝑛 polys , 𝑓 𝑛 polys ) . The image of the projection of I onto the last coordinate is the ideal, h 𝑓 , . . . , 𝑓 𝑛 polys i .A syzygy of I is an element z = ( 𝛼, 𝛼 ) such that 𝛼 = . The setof all syzygies of I is denoted by Syz (I) , it is an 𝐴 -module.A monomial of M is an element of the form 𝜇 e 𝑖 , with 𝜇 ∈ Mon ( 𝐴 ) and 𝑖 ∈ È , 𝑛 polys É . A term of M is an element of the form 𝑐 m where 𝑐 ∈ 𝑅 and m is a monomial of M . As before, the set of terms(resp. monomials) of M is denoted by Ter ( M ) (resp. Mon ( M ) ).A monomial ordering on M is an ordering ă on Mon ( M ) with(1) if m ă n , then 𝜇 m ă 𝜇 n ;(2) if 𝜇 < 𝜈 , then 𝜇 m ă 𝜈 m .Examples of orderings on M are the position over term (or PoT)ordering, defined as 𝜇 e 𝑖 ă PoT 𝜈 e 𝑗 if 𝑖 < 𝑗 , or 𝑖 = 𝑗 and 𝜇 < 𝜈 , andthe term over position (or ToP) ordering, defined as 𝜇 e 𝑖 ă ToP 𝜈 e 𝑗 if 𝜇 < 𝜈 , or 𝜇 = 𝜈 and 𝑖 < 𝑗 .As in the case of polynomials, a monomial ordering on M can beextended into a partial term ordering. Let s = 𝑎𝜇 e 𝑖 and t = 𝑏𝜈 e 𝑗 ∈ Ter ( M ) , we write s ≃ t if s and t are incomparable, that is, if 𝜇 = 𝜈 and 𝑖 = 𝑗 . We write s = t if 𝑎 = 𝑏 , 𝜇 = 𝜈 and 𝑖 = 𝑗 .We say that s ĺ t if s ă t or s ≃ t , and similarly, s ň t impliesthat s ; t . It is harmless because ≃ is an equivalence relation and ă is a total order on the quotient, so, for example, if s ≃ t and s ă u , then t ă u .Given an element p = ( 𝛼, 𝛼 ) ∈ I , we define the leading term lt ,leading monomial lm and leading coefficient lc of p to be those of 𝛼 . The signature of p is the leading term of the module element 𝛼 for the module monomial ordering ă , i.e. , the largest module termappearing in 𝛼 , and it is denoted as sig ( p ) . In this section, we introduce generalizations to rings of construc-tions used in signature Gröbner bases over fields. These construc-tions extend those introduced in [12].The key idea, as in the case of fields, is that for each element f = ( 𝛼, 𝛼 ) , we need only keep track of sig ( f ) = lt ( 𝛼 ) and 𝛼 , insteadof the full module representation 𝛼 . For that purpose, we restrictto operations which do not cancel the signatures. Definition 2.1.
Let f , g ∈ I . The sum f + g is called regular if sig ( f ) ; sig ( g ) , and singular if sig ( f ) = − sig ( g ) . The nature of the operation yields information about the signa-ture of the result, as follows.
Proposition 2.2.
Let f = ( 𝛼, 𝛼 ) and g = ( 𝛽, 𝛽 ) ∈ I , let h = ( 𝛾,𝛾 ) = f + g . Then, • f + g is a regular addition iff sig ( h ) = max ( sig ( f ) , sig ( g )) ; • f + g is a non-singular addition iff sig ( h ) = sig ( f ) + sig ( g ) ≃ sig ( f ) ≃ sig ( g ) ; • f + g is a singular addition iff sig ( h ) ň sig ( f ) ≃ sig ( g ) . The proof of the proposition is straightforward. Note that in asingular addition, the signature of the result cannot be computedfrom the signatures of the summands. This phenomenon is called n Two Signature Variants Of Buchberger’s AlgorithmOver Principal Ideal Domains Conference’21, July 2021, Washington, DC, USA a signature drop , and signature-based algorithms must disallow sig-nature drops, and thus singular operations, in order to keep trackof the signatures.Signature Gröbner bases, as in the case of fields, are charac-terized by the fact that all elements of the ideal are s-reducible,namely, reducible without increasing the signature. In the case ofrings, different notions of reduction exist, namely weak and strongreductions, as well as modular reductions by the coefficients. Inthis paper, we only consider strong reductions, which require thatthe leading coefficient of the reducer divides that of the reducee.Those reductions allow to define strong Gröbner bases. In the restof the paper, we shall omit the “strong” qualificative. Definition 2.3.
Let
G ⊂ I and f , h ∈ I .We say that f (strongly) s-reduces to h modulo G if there exists g 𝑖 ∈ G and 𝑡 𝑖 ∈ Ter ( 𝐴 ) such that(1) lt ( f ) = 𝑡 𝑖 lt ( g 𝑖 ) (2) h = f − 𝑡 𝑖 g 𝑖 (3) 𝑡 𝑖 sig ( g 𝑖 ) ĺ sig ( f ) If the signature inequality is strict, 𝑡 𝑖 sig ( g 𝑖 ) ň sig ( f ) , it is called a regular s -reduction, and if 𝑡 𝑖 sig ( g 𝑖 ) = sig ( f ) , it is called a singulars -reduction.By abuse of language, we extend these definitions to sequences ofreductions. We say that f s-reduces (resp. regular s-reduces) to zeroif there exists a sequence of s-reductions (resp. regular s-reductions)whose final result has polynomial part equal to 0. Remark 2.4. If f s -reduces to h modulo G , then sig ( h ) ĺ sig ( g ) ,with equality iff the reduction is regular and strict inequality iff thereduction is singular. Note that an s-reduction might be neither reg-ular nor singular, in which case sig ( h ) ≃ sig ( f ) . We then recall the definition of (strong) signature Gröbner bases. Definition 2.5.
Let
G ⊂ I and T ∈ Ter ( M ) . G is called a(strong) signature Gröbner basis (or Sig-GB for short) up to signature T if every f ∈ I with sig ( f ) ň T is s -reducible modulo G . G is calleda signature Gröbner basis if it is a signature Gröbner basis up to T for all T . The original motivation for the use of signatures is to maintaina list of signatures of known syzygies, and use it to predict reduc-tions to zero. Additionally, the last coordinates of elements of aSig-GB form a GB in the classical sense. The proof of that fact [6,Lem. 4.6] can be directly extended to rings. So signature-based al-gorithms allow to compute classical Gröbner bases in a more effi-cient way.This use of syzygies applies to our case as well, and requires todefine reductions by signatures of syzygies.
Definition 2.6.
Let G 𝑧 ⊂ Syz (I) and let f ∈ I , with sig ( f ) = 𝑎𝜇 e 𝑖 , for 𝑎 ∈ 𝑅 and 𝜇 ∈ Mon ( 𝐴 ) . We say that f is sig-reducible modulo G 𝑧 if there exists z ∈ G 𝑧 such that sig ( z ) divides sig ( f ) .Let T ∈ Mon ( M ) , we say that G 𝑧 is a Sig-basis of syzygies (resp.basis up to T ) if any syzygy of I (resp. syzygy with signature ň T )is sig-reducible by G 𝑧 . In [14], a signature GB is called a strong GB. We use Sig-GB here to avoid conflictwith the existing notion of strong GB over rings.
Proposition 2.7.
Let G be a Sig-GB up to signature T , G 𝑧 ⊂ Syz (I) and f ∈ I with sig ( f ) ĺ T . If f is sig-reducible modulo G 𝑧 ,then f regular s-reduces to 0 modulo G . Proof.
Let z ∈ G 𝑧 be such that there exists 𝑡 ∈ Ter ( 𝐴 ) with 𝑡 sig ( z ) = sig ( f ) . Let g = f − 𝑡 z , it has signature ň T so it s-reducesto 0, and since its lt is equal to that of f , f regular reduces to 0. (cid:3) In the classical case, without signatures, it is sometimes conve-nient to consider expanded sequences of reductions, leading to thenotion of standard representation. With signatures, it turns outthat a natural generalization of that notion encompasses both s-reductions and sig-reductions.
Definition 2.8.
Let G = { g , . . . , g 𝑟 } ⊂ I , G 𝑧 = { z , . . . , z 𝑠 } ⊂ Syz (I) and h ∈ I . Let 𝑡 ( ) 𝑢 ∈ Ter ( 𝐴 ) , 𝑖 𝑢 ∈ È , 𝑟 É , where 𝑢 ∈ È , 𝑘 É and 𝑘 ∈ N , 𝑡 ( ) 𝑣 ∈ Ter ( 𝐴 ) , 𝑗 𝑣 ∈ È , 𝑠 É , where 𝑣 ∈ È , 𝑙 É and 𝑙 ∈ N ,be such that the equality h = Í 𝑘𝑢 = 𝑡 ( ) 𝑢 g 𝑖 𝑢 + Í 𝑙𝑣 = 𝑡 ( ) 𝑣 z 𝑗 𝑣 (1) holds in I , with(1) lt ( 𝑡 ( ) g 𝑖 ) > lt ( 𝑡 ( ) g 𝑖 ) ≥ lt ( 𝑡 ( ) g 𝑖 ) ≥ · · · ≥ lt ( 𝑡 ( ) 𝑘 g 𝑖 𝑘 ) ;(2) for all 𝑢 ∈ È , 𝑘 É , sig ( 𝑡 ( ) 𝑢 g 𝑖 𝑢 ) ĺ sig ( h ) ;(3) sig ( 𝑡 ( ) z 𝑗 ) ŋ sig ( 𝑡 ( ) z 𝑗 ) ľ sig ( 𝑡 ( ) z 𝑗 ) ľ . . . ľ sig ( 𝑡 ( ) 𝑙 z 𝑗 𝑙 ) ;(4) for all 𝑣 ∈ È , 𝑙 É , sig ( 𝑡 ( ) 𝑣 z 𝑗 𝑣 ) ĺ sig ( h ) .If such a decomposition exists, we say that (2.8) is a standard Sig-representation of h with respect to (G , G 𝑧 ) . Proposition 2.9.
Let
G ⊂ I , G 𝑧 ⊂ Syz (I) such that everyelement of I admits a standard Sig-representation by (G , G 𝑧 ) . Then G is a Sig-GB and G 𝑧 is a Sig-basis of syzygies. Proof.
Let h ∈ I . By assumption it admits a standard Sig-representation as in (2.8). If h is not a syzygy, then by property 1on the leading terms, lt ( h ) = 𝑡 ( ) lt ( g 𝑖 ) , and by property 2 on thesignatures, this is an s-reduction of h . If h is a syzygy, then againby the property on the leading terms, 𝑘 = , and properties 3 and4 on the signatures imply that h is sig-reducible by z 𝑗 . (cid:3) We recall how S-polynomials are defined with signatures. First,we give some definitions associated with pairs of module elements.
Definition 2.10.
Let f , g ∈ I , 𝑡 f = lcmlt ( f , g ) lt ( f ) , 𝑡 g = lcmlt ( f , g ) lt ( g ) .The term degree of the pair ( f , g ) is tdeg ( f , g ) = lcmlt ( f , g ) = 𝑡 f lt ( f ) = 𝑡 g lt ( g ) . The monomial degree mdeg ( f , g ) of the pair ( f , g ) is the monomial part of the term degree.The pair ( f , g ) is called regular if 𝑡 f sig ( f ) ; 𝑡 g sig ( g ) and it iscalled singular if 𝑡 f sig ( f ) + 𝑡 g sig ( g ) = . The signature of the pair ( f , g ) is sig ( f , g ) = max ( 𝑡 f sig ( f ) , − 𝑡 g sig ( g )) . Definition 2.11.
Let f and g ∈ I . The S-polynomial of f and g is S-Pol ( f , g ) = lcmlt ( f , g ) lt ( f ) f − lcmlt ( f , g ) lt ( g ) g . Remark 2.12.
Let h = S-Pol ( f , g ) . Then sig ( h ) ĺ sig ( f , g ) , withequality iff the pair ( f , g ) is regular, and strict inequality iff it is sin-gular. onference’21, July 2021, Washington, DC, USA Maria Francis and Thibaut Verron In order to ensure that elements are strongly s-reducible modulo G , we need to compute G-polynomials . The G-polynomial of 𝑓 and 𝑓 is a polynomial 𝑓 such that any linear combination of 𝑓 and 𝑓 not cancelling the leading terms is reducible by 𝑓 . It is definedby using Bézout relations to make the leading coefficient as smallas possible. Definition 2.13.
Let f and g ∈ I . Let 𝑢, 𝑣 be Bézout coefficientsof lc ( f ) and lc ( g ) , that is, 𝑢 lc ( f ) + 𝑣 lc ( g ) = gcd ( lc ( f ) , lc ( g )) . TheG-polynomial of f and g associated to ( 𝑢, 𝑣 ) is defined as G-Pol 𝑢,𝑣 ( f , g ) = 𝑢 lcmlm ( f , g ) lm ( f ) f + 𝑣 lcmlm ( f , g ) lm ( g ) g . The coefficients 𝑢 and 𝑣 are not uniquely determined, and wecan use this fact to ensure that G-polynomials never represent asingular operation. Proposition 2.14.
Let f and g ∈ I . Then there exists 𝑢, 𝑣 suchthat sig ( G-Pol 𝑢,𝑣 ( f , g )) ≃ sig ( f , g ) . Proof.
If the pair is regular, there is nothing to prove, and anypair of Bézout coefficients works. Otherwise, let 𝑎 = lc ( f ) , 𝑏 = lc ( g ) , 𝑐 = lc ( sig ( f )) , 𝑑 = lc ( sig ( g )) , and 𝑔 = gcd ( 𝑎,𝑏 ) . We want toprove that there exists 𝑢, 𝑣 such that 𝑎𝑢 + 𝑏𝑣 = 𝑔 and 𝑐𝑢 + 𝑑𝑣 ≠ . If 𝑎𝑑 − 𝑏𝑐 = , 𝑎 ( 𝑐𝑢 + 𝑑𝑣 ) = 𝑐 ( 𝑎𝑢 + 𝑏𝑣 ) ≠ , so again any pair of Bézoutcoefficients works. Otherwise, assume that 𝑎𝑢 + 𝑏𝑣 = , 𝑐𝑢 + 𝑑𝑣 = ,and consider the pair 𝑢 ′ = 𝑢 + 𝑏 , 𝑣 ′ = 𝑣 − 𝑎 . Then 𝑎𝑢 ′ + 𝑏𝑣 ′ = 𝑔 and 𝑐𝑢 ′ + 𝑑𝑣 ′ = 𝑏𝑐 − 𝑎𝑑 ≠ . (cid:3) Remark 2.15.
With the notations of the proof,
G-Pol 𝑢 ′ ,𝑣 ′ ( f , g ) = G-Pol 𝑢,𝑣 ( f , g ) − S-Pol ( f , g ) . In practice, we shall always consider such a pair of Bézout coeffi-cients, and call the corresponding G-polynomial the
G-polynomialof f and g , denoted by G-Pol ( f , g ) . Note that sig ( G-Pol ( f , g )) ≃ sig ( f , g ) and lm ( G-Pol ( f , g )) = mdeg ( f , g ) . Definition 2.16.
Let
G ⊂ I . We say that G is complete if everyG-polynomial of elements of G is s-reducible modulo G . It is always possible to make G complete by adding G-polynomialsto G until the property holds. We use a similar process, but on thesignatures, to create syzygies with small signature coefficients. Definition 2.17.
Let z , z ∈ Syz (I) with respective signatures 𝑎 𝑘 𝜇 𝑘 e 𝑖 , 𝑘 = , , sharing the same index 𝑖 . Let 𝑑 = gcd ( 𝑎 , 𝑎 ) ,and let 𝑢 , 𝑢 be Bézout coefficients. Let 𝜇 = lcm ( 𝜇 , 𝜇 ) . The sigG -combination of z and z is defined as sigG-Comb ( z , z ) = 𝑢 𝜇𝜇 z + 𝑢 𝜇𝜇 z , and its signature is 𝑑𝜇 e 𝑖 . Note that contrary to the case of polynomi-als, the result does not depend on the choice of 𝑢 and 𝑢 .Let G 𝑧 ⊂ Syz (I) , we say that G 𝑧 is sigG -complete if any sigG -combination z of elements of G 𝑧 is sig-reducible by G 𝑧 . We conclude this section with a few definitions which will giveuseful criteria to prove the correctness and for detecting uselesssyzygies in Algorithm 1 given below. These constructions are adapted Terminology and notations vary: this construction is called T-polynomial in [20], 𝑆𝐿 in [22], CP2 critical pairs in [16], S-polynomial of type 1 in [17], G-polynomial in [2]and GCD-polynomial in [7, 8]. from the ones defined for the GVW algorithm over fields [13]. Thefirst definition is that of super reducible elements . Definition 2.18.
Let
G ⊂ I , and f ∈ I . f is super reducible by G if there exists g ∈ G and 𝑡 ∈ Ter ( 𝐴 ) such that sig ( f ) = 𝑡 sig ( g ) and lm ( 𝑡 g ) = lm ( f ) . Note that unlike in the case of fields, we do not require that asuper reduction is a reduction. However, under some hypotheses,an element which is super reducible is necessarily s-reducible.
Proposition 2.19.
Let
G ⊂ I be complete, and let f ∈ I . If f issuper reducible by G , then f is s -reducible by G . Proof.
Assume for a contradiction that f is super reducible by G , not s-reducible, and has minimal signature for this property. Let g be such that there exists 𝑡 ∈ Ter ( 𝐴 ) with sig ( f ) = 𝑡 sig ( g ) and lm ( 𝑡 g ) = lm ( f ) . If lt ( 𝑡 g ) = lt ( f ) , then g is a s-reducer of f . Otherwise, lm ( f − 𝑡 g ) = lm ( f ) . Since sig ( f − 𝑡 g ) ň sig ( f ) , byminimality of sig ( f ) , f − 𝑡 g is s-reducible modulo G . Let g be sucha reducer, with lt ( f − 𝑡 g ) = 𝑡 g , and so lt ( f ) = 𝑡 lt ( g )+ 𝑡 lt ( g ) .The signature satisfies 𝑡 sig ( g ) ĺ sig ( f − 𝑡 g ) ň sig ( 𝑡 g ) . Let g = G-Pol ( g , g ) , by definition of the G-pol there exists 𝑡 ∈ Ter ( 𝐴 ) such that 𝑡 lt ( g ) = lt ( f ) , and 𝑡 sig ( g ) ≃ 𝑡 sig ( g ) ≃ sig ( f ) . By hypothesis g is s-reducible by G , and a s-reducer of g is a s-reducer of f . (cid:3) The last definition is that of the covered property.
Definition 2.20.
Let ( f , f ) ∈ I be a pair. Let G ⊂ I and G 𝑧 ⊂ Syz (I) . The pair ( f , f ) is covered by (G , G 𝑧 ) if there exists g ∈ G , z ∈ G 𝑧 , 𝑡, 𝑡 ( 𝑧 ) ∈ Ter ( 𝐴 ) such that • if 𝑡 ≠ , sig ( f , g ) ≃ 𝑡 sig ( g ) ; • if 𝑡 ( 𝑧 ) ≠ , sig ( f , g ) ≃ 𝑡 ( 𝑧 ) sig ( z ) ; • sig ( f , g ) = 𝑡 sig ( g ) + 𝑡 ( 𝑧 ) sig ( z ) ; • lm ( 𝑡 g ) < mdeg ( f , g ) . This cover criterion looks more complicated to implement thanin the case of fields, due to the need to consider linear combina-tions. However, one can use sigG-combinations of elements of G and elements of G 𝑧 to compute elements with signature as smallas possible and same leading monomial, and reduce the cover testto a single divisibility test. The first algorithm which we present in this paper is a signature-enabled version of Kandry-Rody and Kapur’s algorithm. The algo-rithm works similarly to Buchberger’s algorithm, but adds both S-and G-polynomials to the basis. The signature variant follows theconstruction of the GVW algorithm [13, 14].The correctness of the algorithm is stated by the following the-orem (proved in Section 3.2), and adapted from [14, Thm. 2.4].
Theorem 3.1.
Let
G ⊂ I complete , G 𝑧 ⊂ Syz (I) sigG -complete,such that for all signatures T , there exists some g ∈ G ∪ G 𝑧 such that In the paper [12], the notion was called 1-singular reducible. n Two Signature Variants Of Buchberger’s AlgorithmOver Principal Ideal Domains Conference’21, July 2021, Washington, DC, USA sig ( g ) divides T . Assume that every regular pair of elements of G iscovered by (G , G 𝑧 ) . Then, G is a Sig-Gröbner basis and G 𝑧 is a Sig-basis of syzygies. The algorithm ensures that all the assumptions of the theoremhold: • G is complete and G 𝑧 is sigG-complete because G-polynomialsare added to the queue of pairs to be reduced for additioninto G , and sigG-combinations to G 𝑧 ; • there exists, for all T , a g with sig ( g ) dividing sig ( T ) , be-cause we process all elements ( e 𝑖 , 𝑓 𝑖 ) , thus ensuring thatthere is an element with signature e 𝑖 in either G or G 𝑧 forall 𝑖 ; • every regular pair is covered by (G , G 𝑧 ) because, for eachregular pair, we compute the corresponding S-polynomial,reduce it and add the result to the basis, thus creating a cov-ering element for the pair.The resulting algorithm is described in Algorithm 1. Note thatfor each element f = ( 𝛼, 𝛼 ) ∈ I , we only keep track of sig ( f ) = lt ( 𝛼 ) and 𝛼 . The routines SigReduce and
RegularReduce computethe sig-reduction of a signature by a basis of syzygies (the resultbeing either or the signature itself), and the regular reduction ofan element of I by a Sig-GB, respectively.A technical point is that the theorem allows to eliminate S-polynomialsobtained from a pair which is covered, but not necessarily G-polynomials.This requires to keep track of how each element was computed. Inthe pseudo-code algorithm, we do it by keeping for each new ele-ment its so-called type , which can take three values: N , indicatinga polynomial from the input; S ( 𝑖, 𝑗 ) , indicating the S-polynomialof g 𝑖 and g 𝑗 ; and G ( 𝑖, 𝑗 ) , indicating the G-polynomial of g 𝑖 and g 𝑗 .On top of that, we add some tests to eliminate some G-polynomials.Firstly, if lc ( g 𝑖 ) divides lc ( g 𝑗 ) , then one can choose the Bézout co-efficients such that G-Pol ( g 𝑖 , g 𝑗 ) is a multiple of g 𝑖 , and thus it isautomatically s-reducible modulo G .Secondly, thanks to Proposition 2.7, we know that we can imme-diately disregard any element whose signature is divisible by thatof a syzygy. This partially extends the cover criterion to G-polys.Thirdly, note that we cannot use Proposition 2.19 to eliminate G-polynomials which would be super reducible: indeed, that proposi-tion requires that G be complete, and the G-polynomial being pro-cessed might be necessary for that. Furthermore, the G-polynomialG-Pol ( g 𝑖 , g 𝑗 ) is always super reducible by at least one of g 𝑖 , g 𝑗 .However, if f ∈ I is both super reducible and s-reducible by g ∈ G , that is, if there exists 𝑡 ∈ Ter ( 𝐴 ) such that 𝑡 sig ( g ) = sig ( f ) and 𝑡 lt ( g ) = lt ( f ) , then the proof of Proposition 2.19 shows that f is s-reducible by G , without any hypothesis of completeness. Thussuch an element can be immediately discarded.For some signature orderings, it is also possible to predict inadvance the signature of some syzygies, with the F5 criterion. Thiscriterion can be implemented in our setting exactly as in the caseof fields, for instance by adding signatures to G 𝑧 , and we do notdetail it here. The proof of Theorem 3.1 is adapted from the proof of [14, Thm. 2.4].
Proof of Th. 3.1.
We prove the implication by contradiction.Assume that there exists f ∈ I such that f is not s-reducible mod-ulo G and f is not sig-reducible modulo G 𝑧 , and pick f with minimalsignature T for this property. Let g ∈ G , z ∈ G 𝑧 , 𝑡, 𝑡 𝑧 ∈ Ter ( 𝐴 ) such that T = 𝑡 sig ( g ) + 𝑡 𝑧 sig ( z ) and such that lm ( 𝑡 g ) is mini-mal for that property. By hypothesis, such a decomposition exists(with either 𝑡 = or 𝑡 𝑧 = ).First, we prove that 𝑡 g is not regular s-reducible modulo G . In-deed, if it were, let g be a regular reducer of 𝑡 g . Consider the pair ( g , g ) , let 𝜇 = lcmlm ( g , g ) and let 𝜎 = sig ( g , g ) . By proper-ties of the lcm, there exist some terms 𝑡 , 𝑡 such that 𝜇 = lm ( 𝑡 g ) , 𝜎 = 𝑡 sig ( g ) , and 𝑡 𝑡 = 𝑡 . Furthermore, since g is a regular re-ducer of 𝑡 g , the pair ( g , g ) is regular ([14, Lemma. 2.3]).By assumption, the pair is covered by (G , G 𝑧 ) , so there exists g ∈ G , z ∈ G 𝑧 , 𝑡 , 𝑡 𝑧 ∈ Ter ( 𝐴 ) , such that 𝜎 = 𝑡 sig ( g ) + 𝑡 𝑧 sig ( z ) , and lm ( 𝑡 g ) < 𝜇 . So all in all, T = 𝑡 𝑡 sig ( g ) + 𝑡 𝑧 sig ( z ) + 𝑡 𝑧 sig ( z ) . and lm ( 𝑡 𝑡 g ) < lm ( 𝑡 𝑡 g ) = lm ( 𝑡 g ) .Let z be sigG-Comb ( z , z ) , its signature divides the sum 𝑡 𝑧 sig ( z )+ 𝑡 𝑧 sig ( z ) , and thus, the existence of ( g , z ) contradicts the min-imality of lm ( 𝑔 ) .So 𝑡 g is not regular s-reducible modulo G . Now we considertwo distinct cases, depending on whether f is a syzygy or not.If f is not a syzygy, lm ( f ) ≠ lm ( 𝑡 g ) , because otherwise f wouldbe super reducible by g and thus, since G is complete, s-reducibleby G . Let f = f − 𝑡 g − 𝑡 𝑧 z , so sig ( f ) ň T , and lt ( f ) = max ( lt ( f ) , 𝑡 lt ( g )) . Since sig ( f ) ň T , by minimality of f , f s-reduces to 0 modulo G . But since lt ( f ) = max ( lt ( f ) , 𝑡 lt ( g )) , anys-reduction of f is a regular reduction of either f or 𝑡 g , which isa contradiction.If f is a syzygy, we proceed similarly, but now lm ( f ) = . So thefact that f is s-reducible implies that 𝑡 g must be regular reducible,which is impossible. So 𝑡 g = and f is sig-reducible by z . (cid:3) The second algorithm which we present is adapted from that ofLichtblau [17], which is itself adapted from that of Pan [22]. Simi-lar to the previous algorithm, this algorithm also adds S- and G-polynomials to the basis, but it tries to limit the growth of thelength of the queue by adding at most one new polynomial for eachpair, either an S- or a G-polynomial, by the following construction.
Definition 4.1.
Let f , g ∈ I . The SG-polynomial of f and g isdefined as: SG-Pol ( f , f ) = ( S-Pol ( f , g ) if lc ( f ) | lc ( g ) or lc ( g ) | lc ( f ) G-Pol ( f , g ) otherwise . Note that compared to the previous algorithm, it leads to com-puting fewer S-polynomials, but not fewer G-polynomials, sinceit is always useless to compute a G-polynomial when one of theleading coefficients divides the other. The correctness of the algo-rithm is ensured by the following theorem, which will be provedin Section 4.2.
Theorem 4.2.
Let
G ⊂ I complete and G 𝑧 ⊂ Syz (I) sigG -complete such that onference’21, July 2021, Washington, DC, USA Maria Francis and Thibaut Verron
Algorithm 1:
Kandri-Rody - Kapur’s algo., with signa-tures
Input : 𝐹 ⊂ 𝐴 Output :
G ⊂
Ter ( M ) × 𝐴 Sig-GB of I , G 𝑧 ⊂ Ter ( M ) × { } Sig-basis of syzygies of I G ← ∅ ; G 𝑧 ← ∅ ; 𝑟 ← ; 𝑄 ← [( e 𝑖 , 𝑓 𝑖 , N ) for 𝑖 ∈ { , . . . , 𝑚 }] ; while 𝑄 is not empty : Take and remove ( s , 𝑓 , type ) from 𝑄 , with s minimal; if ∃ z ∈ G 𝑧 s.t. sig ( z ) divides s : pass ; /* Prop. 2.7 */ elif type is S ( 𝑖, 𝑗 ) and ( g 𝑖 , g 𝑗 ) is covered by (G , G 𝑧 ) : pass ; /* Cover criterion */ else: 𝑔 ← RegularReduce (( s , 𝑓 ) , G) ; if 𝑔 = : Add s to G 𝑧 , together with sigG-combinations; elif ∃ g 𝑖 ∈ G s.t. lt ( g 𝑖 ) s = lt ( 𝑔 ) sig ( g 𝑖 ) : pass ; /* Super and s-reducible */ else: g 𝑟 + ← ( s , 𝑔 ) , add it to G ; for 𝑖 ∈ { , . . . , 𝑟 } : t ← sig ( g 𝑖 , g 𝑟 + ) ; if the pair ( g 𝑖 , g 𝑟 + ) is regular : Add ( t , S-Pol ( g 𝑖 , g 𝑟 + ) , S ( 𝑖, 𝑟 + )) to 𝑄 if none of lc ( g 𝑖 ) , lc ( g 𝑟 + ) divides the other : Add ( t , G-Pol ( g 𝑖 , g 𝑟 + ) , G ( 𝑖, 𝑟 + )) to 𝑄 return G , G 𝑧 Algorithm 2:
Pan/Lichtblau’s algo., with signatures
Input : 𝐹 ⊂ 𝐴 Output :
G ⊂
Ter ( M ) × 𝐴 Sig-GB of I , G 𝑧 ⊂ Ter ( M ) × { } Sig-basis of syzygies of I G ← ∅ ; G 𝑧 ← ∅ ; 𝑟 ← ; 𝑄 ← [( e 𝑖 , 𝑓 𝑖 , N ) for 𝑖 ∈ { , . . . ,𝑚 }] ; while 𝑄 is not empty : Take and remove ( s , 𝑓 , type ) from 𝑄 , with s minimal; if ∃ z ∈ G 𝑧 s.t. sig ( z ) divides s : pass ; /* Prop. 2.7 */ else: 𝑔 ← RegularReduce (( s , 𝑓 ) , G) ; if 𝑔 = : Add s to G 𝑧 , together with sigG-combinations; elif type is G and ∃ g 𝑖 ∈ G s.t. lt ( g 𝑖 ) s = lt ( 𝑔 ) sig ( g 𝑖 ) or type is S and ( s , 𝑔 ) is super reducible by G : pass ; /* Prop. 2.19 */ else: g 𝑟 + ← ( s , 𝑔 ) , add it to G ; for 𝑖 ∈ { , . . . , 𝑟 } : t ← sig ( g 𝑖 , g 𝑟 + ) ; if the pair ( g 𝑖 , g 𝑟 + ) is non-singular andeither of lc ( g 𝑖 ) , lc ( g 𝑟 + ) divides the other : Add ( t , S-Pol ( g 𝑖 , g 𝑟 + ) , S ( 𝑖, 𝑟 + )) to 𝑄 if none of lc ( g 𝑖 ) , lc ( g 𝑟 + ) divides the other : Add ( t , G-Pol ( g 𝑖 , g 𝑟 + ) , G ( 𝑖, 𝑟 + )) to 𝑄 return G , G 𝑧 • ∀ 𝑖 ∈ È , 𝑛 polys É , ( e 𝑖 , 𝑓 𝑖 ) has a standard Sig-rep. w.r.t. (G , G 𝑧 ) ; • any non-singular SG-polynomial of elements of G has a stan-dard Sig-representation w.r.t. (G , G 𝑧 ) .Then G is a Sig-Gröbner basis and G 𝑧 is a Sig-basis of syzygies. The reason why this construction is sufficient is that if lc ( f ) and lc ( g ) do not divide each other, then the S-polynomial of f and g can be expressed in terms of the S-polynomials of f , g and h = G-Pol ( f , g ) . However, it forces us to also compute non-regularS-polynomials as long as they are non-singular: indeed, sig ( f , g ) ≃ sig ( h ) , and sig ( h ) ≃ sig ( f , h ) ≃ sig ( g , h ) because lm ( h ) is equalup to coefficient to the term degree of S-Pol ( f , g ) , S-Pol ( f , h ) andS-Pol ( g , h ) . So any linear combination of S-Pol ( f , h ) and S-Pol ( g , h ) ,which is necessary to recover S-Pol ( f , g ) , will necessarily be non-regular. The theorem implies that even if we accept all non-singularS-polynomials, the algorithm is correct.Allowing non-regular S-polynomials also means that we can-not a priori use the cover criterion in our algorithm: the algorithmwould not ensure that all regular pairs are covered, but rather, onlythose for which the SG-polynomial is actually an S-polynomial. Wecan however eliminate S-polynomials which are super reducible,since Proposition 2.19 ensures that they s-reduce to 0.The rest of the algorithm, including the processing of syzygies,is done in exactly the same way as in Algorithm 1. The proof of Theorem 4.2 is adapted from that of [22] and [17]with the addition of signatures. First, we prove a useful technicallemma.
Lemma 4.3.
Let G = { g , . . . , g 𝑟 } ⊂ I and T ∈ Ter ( M ) suchthat • for all g ∈ I with sig ( g ) ň T , g s -reduces to 0 modulo G ; • for all g 𝑖 , g 𝑗 ∈ G such that SG-Pol ( g 𝑖 , g 𝑗 ) is non-singular and sig ( g 𝑖 , g 𝑗 ) ĺ T , SG-Pol ( g 𝑖 , g 𝑗 ) s -reduces to modulo G .Let g 𝑖 , g 𝑗 ∈ G such that lc ( g 𝑗 ) divides lc ( g 𝑖 ) . Then any (possiblysingular) linear combination of g 𝑖 and g 𝑗 with signature at most T s -reduces to modulo G . Proof.
Let m 𝑖 = lm ( g 𝑖 ) , m 𝑗 = lm ( g 𝑗 ) , and m 𝑖,𝑗 = lcmlm ( g 𝑖 , g 𝑗 ) .Consider a linear combination h = 𝑡 𝑖 g 𝑖 + 𝑡 𝑗 g 𝑗 . Without loss of gen-erality, combining the reductions, we may assume that 𝑡 𝑖 and 𝑡 𝑗 are terms. If lm ( 𝑡 𝑖 g 𝑖 ) ≠ lm ( 𝑡 𝑗 g 𝑗 ) , say lm ( 𝑡 𝑖 g 𝑖 ) > lm ( 𝑡 𝑗 g 𝑗 ) , thereis nothing to prove, as h can be reduced by g 𝑖 , then g 𝑗 .So assume that lm ( 𝑡 𝑖 g 𝑖 ) = lm ( 𝑡 𝑗 g 𝑗 ) . Note that m 𝑖,𝑗 divides thecommon multiple lm ( 𝑡 𝑖 g 𝑖 ) = lm ( 𝑡 𝑗 g 𝑗 ) , say, lm ( 𝑡 𝑖 g 𝑖 ) = 𝑡 ′ m 𝑖,𝑗 . Byassumption on the leading coefficients, there exists 𝑐 ∈ 𝑅 such that lt ( 𝑡 𝑖 g 𝑖 ) = 𝑡 𝑖 𝑡 ′ m 𝑖,𝑗 m 𝑖 lt ( g 𝑖 ) = 𝑡 𝑖 𝑡 ′ 𝑐 m 𝑖,𝑗 m 𝑗 lt ( g 𝑗 ) .If lm ( h ) = lm ( 𝑡 𝑖 𝑔 𝑖 ) , the leading term of h is lt ( h ) = lt ( 𝑡 𝑖 g 𝑖 ) + lt ( 𝑡 𝑗 g 𝑗 ) = ( 𝑡 𝑖 𝑐 + 𝑡 𝑗 ) 𝑡 ′ m 𝑖,𝑗 m 𝑗 lt ( g 𝑗 ) n Two Signature Variants Of Buchberger’s AlgorithmOver Principal Ideal Domains Conference’21, July 2021, Washington, DC, USA and h is reducible by g 𝑗 . This reduction is a s-reduction by con-struction.The remaining case is the case where lt ( h ) < lm ( 𝑡 𝑖 g 𝑖 ) = lm ( 𝑡 𝑗 g 𝑗 ) .In this case, there exists a term 𝑡 such that 𝑡 𝑖 = 𝑡 m 𝑖,𝑗 m 𝑖 and 𝑡 𝑗 = 𝑐𝑡 m 𝑖,𝑗 m 𝑗 ,and h = 𝑡 S-Pol ( g 𝑖 , g 𝑗 ) . If the pair ( g 𝑖 , g 𝑗 ) is non-singular, thenby hypothesis, S-Pol ( g 𝑖 , g 𝑗 ) = SG-Pol ( g 𝑖 , g 𝑗 ) s-reduces to mod-ulo G , and so does h = 𝑡 S-Pol ( g 𝑖 , g 𝑗 ) . If the pair is singular, thensig ( h ) ň 𝑡 sig ( g 𝑖 , g 𝑗 ) ĺ T , and by hypothesis, h s-reduces to modulo G . (cid:3) Proof of Th. 4.2.
Write G = { g , . . . , g 𝑟 } and G 𝑧 = { z , . . . , z 𝑠 } .For all 𝑖, 𝑗 , let c 𝑖 = lc ( g 𝑖 ) , m 𝑖 = lm ( g 𝑖 ) and m 𝑖,𝑗 = lcmlm ( g 𝑖 , g 𝑗 ) .Let h ∈ I with signature T be such that h is not s-reducible mod-ulo G . Assume that T is minimal for this property, and that amongsuch elements of I with signature T , lm ( h ) is minimal.Consider a decomposition of h with respect to G , h = Í 𝑘𝑢 = 𝜏 𝑢 g 𝑖 𝑢 + Í 𝑙𝑣 = 𝑡 ( 𝑧 ) 𝑣 z 𝑗 𝑣 , (2)such that for all 𝑢, 𝑣 , lt ( 𝜏 𝑢 g 𝑖 𝑢 ) ≥ lt ( 𝜏 𝑢 + g 𝑖 𝑢 + ) , max ( 𝜏 𝑢 sig ( g 𝑖 𝑢 )) ĺ T , max ( 𝑡 ( 𝑧 ) 𝑣 sig ( z 𝑗 𝑣 )) ĺ T and 𝑡 ( 𝑧 ) 𝑣 sig ( z 𝑗 𝑣 ) ≥ 𝑡 ( 𝑧 ) 𝑣 + sig ( z 𝑗 𝑣 + ) . Sucha representation exists, by definition of the signature of h and thefirst hypothesis on G . Assume that, among such representation,this one is minimal in the sense that lm ( 𝜏 g 𝑖 ) is minimal, andthe largest 𝑗 such that lm ( 𝜏 𝑗 g 𝑖 𝑗 ) = lm ( 𝜏 g 𝑖 ) is minimal for thisproperty. For all 𝑢 , let 𝜒 𝑢 = lc ( 𝜏 𝑢 ) . Case 1: h is not a syzygy. We want to prove that lm ( 𝜏 g 𝑖 ) > lm ( 𝜏 g 𝑖 ) . It will in particular imply that lt ( h ) = lt ( 𝜏 g 𝑖 ) , andthus that h is s-reducible modulo G . By minimality of lm ( h ) , thiswill prove that h s-reduces to 0 modulo G .In order to reach a contradiction, assume that lm ( 𝜏 g 𝑖 ) = lm ( 𝜏 g 𝑖 ) .By definition of the least common multiplier, there exists 𝑚 ∈ Mon ( 𝐴 ) such that lm ( 𝜏 g 𝑖 ) = lm ( 𝜏 g 𝑖 ) = 𝑚 m 𝑖 ,𝑖 .If c 𝑖 divides c 𝑖 or c 𝑖 divides c 𝑖 , then by Lemma 4.3, andexpanding the s-reductions, 𝜏 g 𝑖 + 𝜏 g 𝑖 admits a standard Sig-representation, which can be substituted in the representation (4.2),contradicting minimality.For the other case, by assumption the G-polynomial of g 𝑖 and g 𝑖 is s-reducible modulo 𝐺 , so there exists g 𝑖 ∈ 𝐺 such thata. c 𝑖 m 𝑖 ,𝑖 is divisible by lt ( g 𝑖 ) , say, 𝑡 ′ lt ( g 𝑖 ) = c 𝑖 m 𝑖 ,𝑖 ;b. 𝑡 ′ sig ( g 𝑖 ) ĺ sig ( g 𝑖 , g 𝑖 ) ;c. c 𝑖 m 𝑖 ,𝑖 is divisible by lt ( g 𝑖 ) , say, 𝑡 ′ lt ( g 𝑖 ) = c 𝑖 m 𝑖 ,𝑖 ;d. 𝑡 ′ sig ( g 𝑖 ) ĺ sig ( g 𝑖 , g 𝑖 ) .In particular, c 𝑖 divides c 𝑖 , say, c 𝑖 = 𝑎 ′ c 𝑖 . So the SG-polynomialof g 𝑖 and g 𝑖 is an S-polynomial, and by Lemma 4.3, it s-reducesto modulo G . So it admits a standard Sig-representationSG-Pol ( g 𝑖 , g 𝑖 ) = m 𝑖 ,𝑖 m 𝑖 g 𝑖 − 𝑎 m 𝑖 ,𝑖 m 𝑖 g 𝑖 = Õ 𝑗 ≥ 𝑡 ( ) 𝑗 g ( ) 𝑖 𝑗 + Õ syz. , where Í syz. is a linear combination of elements of G 𝑧 . Som 𝑖 ,𝑖 m 𝑖 g 𝑖 = 𝑎 m 𝑖 ,𝑖 m 𝑖 g 𝑖 + Õ 𝑗 ≥ 𝑡 ( ) 𝑗 g ( ) 𝑖 𝑗 + Õ syz. , with lt ( 𝑡 ( ) 𝑗 g ( ) 𝑖 𝑗 ) ≤ lt ( S-Pol ( g 𝑖 , g 𝑖 )) < m 𝑖 ,𝑖 . Since m 𝑖 ,𝑖 isdivisible by m 𝑖 , m 𝑖 ,𝑖 is divisible by m 𝑖 ,𝑖 , say, m 𝑖 ,𝑖 = 𝜇 m 𝑖 ,𝑖 . Table 1: Comparison between Algo. 1 and Algo. 2System
Algo 1 Algo. 2Pairs/red./to 0 Time Pairs/red./to 0 Time
Katsura-4
Katsura-5
Cyclic-5
Cyclic-6 𝜏 g 𝑖 = 𝜒 𝑚𝜇 m 𝑖 ,𝑖 m 𝑖 g 𝑖 = 𝜒 𝑧 𝑚𝜇 m 𝑖 ,𝑖 m 𝑖 g 𝑖 + Õ 𝑗 ≥ 𝜒 𝑧 𝑚𝜇 𝑡 ( ) 𝑗 g ( ) 𝑖 𝑗 + Õ syz.and it is a standard representation of 𝜏 g . Furthermore, since thesignature of g 𝑖 is bounded (property b.), it is also a standard Sig-representation.Similarly, there exists 𝑧 , 𝜇 and ( 𝑡 ( ) 𝑗 , 𝑖 𝑗 ) such that 𝜏 g 𝑖 = 𝜒 𝑚𝜇 m 𝑖 ,𝑖 m 𝑖 g 𝑖 = 𝜒 𝑎 𝑚𝜇 m 𝑖 ,𝑖 m 𝑖 g 𝑖 + Õ 𝑗 ≥ 𝜒 𝑎 𝑚𝜇 𝑡 ( ) 𝑗 g ( ) 𝑖 𝑗 + Õ syz.and it is a standard Sig-representation.We can group both representations together, and obtain a stan-dard Sig-representation of 𝜏 g 𝑖 + 𝜏 g 𝑖 𝜏 g 𝑖 + 𝜏 g 𝑖 = 𝜒 𝑎 𝑚𝜇 m 𝑖 ,𝑖 m 𝑖 g 𝑖 + 𝜒 𝑎 𝑚𝜇 m 𝑖 ,𝑖 m 𝑖 g 𝑖 + Õ . . . = (cid:18) 𝜒 𝑐 ′ 𝜇 m 𝑖 ,𝑖 m 𝑖 + 𝜒 𝑎 𝜇 m 𝑖 ,𝑖 m 𝑖 (cid:19) 𝑚 g 𝑖 + Õ . . . = (cid:16) 𝜒 𝑎 + 𝜒 𝑎 (cid:17) m 𝑖 ,𝑖 m 𝑖 𝑚 g 𝑖 + Õ . . . which, substituted into (4.2), contradicts the minimality assump-tion. Case 2: h is a syzygy. The same proof as above, if lt ( h ) = ,implies that 𝑘 = in (4.2). Now, consider, among all decompo-sitions of the form (4.2), one where 𝑗 = max ( 𝑣 : 𝑡 ( 𝑧 ) 𝑣 sig ( z 𝑗 𝑣 )) ≃ 𝑡 ( 𝑧 ) sig ( z 𝑗 ) is minimal. Assume that 𝑗 > , and thus 𝑡 ( 𝑧 ) sig ( z 𝑗 ) ≃ 𝑡 ( 𝑧 ) sig ( z 𝑗 ) . Then, by assumption, the sigG-comb. of z and z is sig-reducible by G 𝑧 , thus, there exists 𝑡 ′ ∈ Ter ( 𝐴 ) and 𝑗 ′ ∈{ , . . . , 𝑠 } such that 𝑡 ′ sig ( z 𝑗 ′ ) = 𝑡 ( 𝑧 ) sig ( z 𝑗 ) + 𝑡 ( 𝑧 ) sig ( z 𝑗 ) , and so z : = 𝑡 ( 𝑧 ) z + 𝑡 ( 𝑧 ) z − 𝑡 ′ sig ( z 𝑗 ′ ) has signature ň T . So subtract-ing z from the decomposition (4.2) results in a decomposition withfewer terms matching the signature of h , contradicting the mini-mality of 𝑗 . (cid:3) In this section, we briefly describe additional criteria which canbe used to eliminate elements. Firstly, in both algorithms, one can onference’21, July 2021, Washington, DC, USA Maria Francis and Thibaut Verron
Table 2: Comparative timings for module computations with the signature-based algorithms and with Magma (in seconds)System
With signatures (Algo. 1) MagmaSig-GB Recons.
Total GB GB with coordinates Module of syzygiesCyclic-5 >24h >24h use Buchberger’s coprime and chain criteria (either as-is or us-ing Gebauer and Möller’s implementation). Buchberger’s criteriondoes not require any modification to work with signatures, whereasthe chain criterion needs to ensure that the signature of the pairsused to discard the redundant one is small enough [15]. Note thatin all cases, we need to consider terms (with their coefficients) andnot just monomials. For Lichtblau’s algorithm, refined versions ofthose criteria relaxing the condition on the coefficients have beendescribed in [17] and can also be used here.We have already stated that some criteria can be used to elim-inate pairs based on their signatures. We have also already men-tioned the idea of the F5 criterion, filling the basis of syzygies withthe signatures of predictable syzygies, as well as the possibility ofdiscarding G-polynomials which are sig-reducible by G 𝑧 , or whichare super reducible and s-reducible by g ∈ G . Similarly, in Licht-blau’s algorithm, one can discard any S-polynomial which is superreducible by G .A natural question is whether the cover criterion would allow tosystematically discard G-polynomials in Algorithm 1, or to discardnew elements in Algorithm 2 (including non-regular S-polynomials).Experimentally, it appears that indeed, most such elements whichare covered can be discarded without impacting the correctness ofthe algorithm.Another point which can have a large impact on the complex-ity is the choice of the order of the pairs, i.e. , how to break tiesbetween elements with signatures which are ≃ . A strategy whichseems to yield good results over Z is to compare the absolute valueof the coefficient of the signatures, so as to create super reducersand covering candidates sooner.We have only mentioned top reductions, namely, reductions ofthe leading coefficient, but as usual, the definitions generalize toallow reductions of the rest of the terms. Finally, we have onlydefined reductions where the leading coefficient of the reducer di-vides the coefficient to be reduced. In some rings, and in particularin the case of Euclidean rings, it is also possible to perform mod-ular reductions on the coefficients without impacting the correct-ness of the result. This significantly improves the performances ofthe algorithm. The same can be done for sig-reductions. We have written a prototype implementation of both algorithmsin
Magma , for the PoT ordering and 𝑅 = Z . We report in Table 1data on the number of pairs being processed, reduced and reducedto zero for different benchmark systems (Katsura- 𝑛 and Cyclic- 𝑛 ),as well as indicative computation times. In practice, it appears thatAlgo. 1 is more efficient than Algo. 2, both in terms of number ofcomputed pairs and in time. This appears to be due to the relaxed https://gitlab.com/thibaut.verron/signature-groebner-rings restrictions allowing non-singular polynomials, more than the lackof criteria.Our prototype implementation of both algorithms of the paperis slower than Magma’s implementation of F4 [9] over Z for merelycomputing Gröbner bases. As mentioned earlier, the computationof signatures also allows to compute the coefficients of the ele-ments of the Gröbner basis in terms of the input, and a basis of themodule of syzygies, by performing and tracking s-reductions [14].The process only depends on the definition of a Sig-GB and a Sig-basis of syzygies, and therefore works in our setting as well. InTable 2, we give the computation time for this reconstruction us-ing Algo. 1, as well as comparable routines in Magma . In severalinstances, we observe that the use of signatures gives a significantspeed-up for those computations.One particularity of signature-based algorithms over rings isthat, due to the partial order on the signatures, they typically com-pute a large number of elements with incomparable signatures.This problem does not appear over fields, and future work will fo-cus on ways to eliminate more of those elements, or to speed-upthe computations at a given signature (for instance using linearalgebra techniques similar to F4). REFERENCES [1] W. Adams and P. Loustaunau.
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