Signature-based algorithms for Gr{ö}bner bases over Tate algebras
SSignature-based algorithmsfor Gröbner bases over Tate algebras
Xavier Caruso
Université de Bordeaux, CNRS,INRIABordeaux, [email protected]
Tristan Vaccon
Université de Limoges; CNRS, XLIMUMR 7252Limoges, [email protected]
Thibaut Verron
Johannes Kepler UniversityInstitute for AlgebraLinz, [email protected]
ABSTRACT
Introduced by Tate in [Ta71], Tate algebras play a major role in thecontext of analytic geometry over the p -adics, where they act as acounterpart to the use of polynomial algebras in classical algebraicgeometry. In [CVV19] the formalism of Gröbner bases over Tatealgebras has been introduced and effectively implemented. Oneof the bottleneck in the algorithms was the time spent on reduc-tion, which are significantly costlier than over polynomials. In thepresent article, we introduce two signature-based Gröbner basesalgorithms for Tate algebras, in order to avoid many reductions.They have been implemented in SageMath. We discuss their su-periority based on numerical evidences. CCS CONCEPTS • Computing methodologies → Algebraic algorithms ; KEYWORDS
Algorithms, Power series, Tate algebra, Gröbner bases, F5 algo-rithm, p -adic precision ACM Reference Format:
Xavier Caruso, Tristan Vaccon, and Thibaut Verron. 2019. Signature-basedalgorithms for Gröbner bases over Tate algebras. In
International Sympo-sium on Symbolic and Algebraic Computation (ISSAC ’19), July 15–18, 2019,Beijing, China.
ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/3326229.3326257
For several decades, many computational questions arising fromgeometry and arithmetics have received much attention, leadingto the development of more and more efficient algorithms and soft-wares. A typical example is the development of the theory of Gröb-ner basis, which provides nowadays quite efficient tools for ma-nipulating ideals in polynomial algebras and, eventually, algebraic
The first author is supported by the French ANR grant CLap–CLap, referenced ANR-18-CE40-0026-01. The third author is supported by the Austrian FWF grant P31571-N32.Permission to make digital or hard copies of all or part 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 cita-tion on the first page. Copyrights for components of this work owned by others thanthe author(s) must be honored. Abstracting with credit is permitted. To copy other-wise, or republish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee. Request permissions from [email protected].
ISSAC ’19, July 15–18, 2019, Beijing, China © 2019 Copyright held by the owner/author(s). Publication rights licensed to the As-sociation for Computing Machinery.ACM ISBN 978-1-4503-6084-5/19/07...$15.00https://doi.org/10.1145/3326229.3326257 varieties and schemes. At the intersection of geometry and num-ber theory, one finds p -adic geometry and, more precisely, the no-tion of p -adic analytic varieties first defined by Tate in 1971 [Ta71],which plays quite an important role in many modern theories andachievements ( e.g. p -adic cohomologies, p -adic modular forms).The main algebraic objects upon which Tate’s geometry is builtare Tate algebras and their ideals. In an earlier paper [CVV19],the authors started to study computational aspects related to Tatealgebras: they introduce Gröbner bases in this context and designtwo algorithms (adapted from Buchberger’s algorithm and the F4algorithm, respectively) for computing them.In the classical setting, the main complexity bottleneck in Gröb-ner basis computations is the time spent reducing elements modulothe basis. The most costly reductions are typically reductions to 0,because they require successively eliminating all terms from thepolynomial; yet their output has little value for the rest of the al-gorithm. Fortunately, it turns out that many such reductions can bepredicted in advance (for example those coming from the obviousequality f д − дf =
0) by keeping track of some information on themodule representation of elements of an ideal, called their signa-ture . This idea was first presented in Algorithm F5 [Fa02] and ledto the development of many algorithms showing different ways todefine signatures, to use them or to compute them. The interestedreader can look at [EF17] for an extensive survey.The Tate setting is not an exception to the wisdom that reduc-tions are expensive. The situation is actually even worse since re-ductions to 0 are theorically the result of an infinite sequence of re-duction steps converging to
0. In practice, the process actually stopsbecause we are working at finite precision; however, the higherthe precision is, the more expensive the reductions to 0 are, for nobenefit. This observation motivates investigating the possibility ofadding signatures to Gröbner basis algorithms for Tate series.
Our contribution.
In this paper, we present two signature-based al-gorithms for the computation of Gröbner bases over Tate algebras.They differ in that they use different orderings on the signatures.Our first variant, called the PoTe (position over term) algorithm,is directly adapted from the G2V algorithm [GGV10]. It adoptsan incremental point of view and uses the so-called cover crite-rion [GVW16] to detect reductions to 0. A key difficulty in the Tatesetting is that the usual way to handle signatures assumes the con-stant term 1 is the smallest one. However, the assumption fails inthe Tate setting. We solve this issue by importing ideas from [L+18]in which the case of local algebras is addressed.In the classical setting, incremental algorithms have the disad-vantage of sometimes computing larger Gröbner bases for inter-mediate ideals, only to discard them later on. In order to mitigate a r X i v : . [ c s . S C ] F e b his misfeature, the F5 algorithm uses a signature ordering tak-ing into account the degree of the polynomials first, in order toprocess lower-degree elements first. In the Tate setting, the de-gree no longer makes sense and a better measure of progressionof the algorithms is the valuation. Nonetheless, similarly to theclassical setting, an incremental algorithm could perform interme-diate computations to high valuation and just discard them lateron. The second algorithm we will present, called the VaPoTe (val-uation over position over term) algorithm, uses an analogous ideato that of F5 to mitigate this problem. Organization of the article.
In §2, we recall the basic definitions andproperties of Tate algebras and Gröbner basis over them, togetherwith the principles of the G2V algorithm. The two next sections aredevoted to the PoTe and the VaPoTe algorithms respectively: theyare presented and their correctness and termination are proved.Finally, implementation, benchmarks and possible future improve-ments, are discussed in §5.
Notations.
Throughout this article, we fix a positive integer n anduse the short notation X for ( X , · · · , X n ) . Given i = ( i , . . . , i n ) ∈ N n , we shall write X i for X i · · · X i n n . In this section, we present the two main ingredients we are go-ing to mix together later on. They are, first, the G2V [GGV10] andGVW [GVW16] signature-based algorithms, and, second, the Tatealgebras and the theory of Gröbner bases over them as developedin [CVV19].
In what follows, we present the G2V algorithm which was de-signed by Gao, Guan and Volny IV in [GGV10] as an incremen-tal variant of the classical F5 algorithm. Our presentation includesthe cover criterion which was formulated later on in [GVW16] byGao, Volny IV and Wang. The incremental point of view is neededfor the application we will discuss in §4. Moreover we believe thatit has two extra advantages: first, it leads to simplified notationsand, more importantly, it shows clearly where intermediate inter-reductions are possible.Let k be a field and k [ X ] denote the ring of polynomials over k with indeterminates X . We endow k [ X ] with a fixed monomialorder ≤ ω . Let I be an ideal in k [ X ] . Let G be a Gröbner basis of I with respect to ≤ ω . Let f ∈ k [ X ] . We aim at computing a GB of theideal I = I + ⟨ f ⟩ . Let M ⊂ k [ X ] × k [ X ] be the k [ X ] -sub-moduledefined by the ( u , v ) such that u f − v ∈ I . The leading monomialof u is the signature of ( u , v ) . Definition 2.1 (Regular reduction) . Let p = ( u , v ) and p = ( u , v ) be in M . We say that p is top-reducible by p if(1) either v = LM ( u ) divides LM ( u ) ,(2) or v v (cid:44) LM ( v ) divides LM ( v ) and: LM ( v ) LM ( v ) · LM ( u ) ≤ LM ( u ) . The corresponding top-reduction is p = p − tp = ( u − tu , v − tv ) where t = LM ( u ) LM ( u ) is the first case and t = LM ( v ) LM ( v ) in the secondcase. This top-reduction is called regular when LM ( u ) > tLM ( u ) ,that is when the signature of the reduced pair p agrees with thatof p ; it is called super otherwise. Definition 2.2 (Strong Gröbner bases) . A finite subset G of M iscalled a strong Gröbner basis (SGB, for short) of M if any nonzero ( u , v ) ∈ M is top-reducible by some element of G .The G2V strategy derives the computation of a Gröbner basisthrough the computation of an SGB. They are related through thefollowing proposition.Proposition 2.3. Suppose that G = {( u , v ) , . . . , ( u s , v s )} is anSGB of M . Then:(1) { u s.t. ( u , ) ∈ G } is a Gröbner basis of ( I : f ) . (2) { v s.t. ( u , v ) ∈ G for some u } is a Gröbner basis of I . To compute an SGB, we rely on J-pairs instead of S-polynomials.
Definition 2.4 (J-pair) . Let p = ( u , v ) and p = ( u , v ) be twoelements in M such that v v (cid:44)
0. Let t = lcm ( LM ( v ) , LM ( v )) and set t i = t / LM ( v i ) for i ∈ { , } . Then: • if LM ( t u ) < LM ( t u ) , the J-pair of ( p , p ) is t p , • if LM ( t u ) > LM ( t u ) , the J-pair of ( p , p ) is t p , • if LM ( t u ) = LM ( t u ) , the J-pair of ( p , p ) is not defined. Definition 2.5 (Cover) . We say that p = ( u , v ) is covered by G ⊂ M if there is a pair ( u i , v i ) ∈ G such that LM ( u i ) divides LM ( u ) and: LM ( u i ) LM ( u ) · LM ( v i ) < LM ( v ) . Theorem 2.6 (Cover Theorem).
Let G be a finite subset of M such that: • G contains ( , f ) ; • the set { д ∈ k [ X ] : ( , д ) ∈ G } forms a Gröbner basis of I .Then G is an SGB of M iff every J-pair of G is covered by G . This theorem leads naturally to the G2V algorithm (see [GGV10,Fig. 1]) which is rephased hereafter in Algorithm 1 (page 4). Weunderline that, in Algorithm 1, the SGB does not entirely appear.Indeed, we remark that one can always work with pairs ( LM ( u ) , v ) in place of ( u , v ) , reducing then drastically the memory occupa-tion and the complexity. The algorithm maintains two lists G and S which are related to the SGB in construction as follows: G ∪( S ×{ }) is equal to the set of all ( LM ( u ) , v ) when ( u , v ) runs over the SGB.The criterion coming from the cover theorem is implemented onlines 10 and 11: the first (resp. the second) statement checks if ( u , v ) is covered by an element of G (resp. an element of S × { } ). Syzygies.
The G2V algorithm does not give direct access to themodule of syzygies of the ideal. However, it does give access toa GB of ( I : f ) (see Proposition 2.3), from which one can recoverpartial information about the syzygies, as shown below. Definition 2.7.
Given f , . . . , f m ∈ k [ X ] , we define Syz ( f , . . . , f m ) = (cid:110) ( a , . . . , a m ) ∈ k [ X ] m s.t. m (cid:213) i = a i f i = (cid:111) . emma 2.8. Let f , . . . , f m generating I and let u , . . . , u s gen-erating ( I : f ) . For i ∈ { , . . . , s } , we write − u i f = a i , f + · · · + a i , m f m ( a i , j ∈ k [ X ]) and define z i = ( a i , , . . . , a i , m , u i ) ∈ Syz ( f , . . . , f m , f ) . Then Syz ( f , . . . , f m , f ) = ( Syz ( f , . . . , f m ) × { }) + ⟨ z , . . . , z s ⟩ . Proof. Let ( a , . . . , a m , u ) ∈ Syz ( f , . . . , f m , f ) . Then u ∈ ( I : f ) and we can write u = (cid:205) si = b i u i . Then the syzygy ( a , . . . , a m , u )− (cid:205) si = b i z i has its last coordinate equal to 0 and thus belongs to ( Syz ( f , . . . , f m ) × { }) , which is enough to conclude. □ Definitions.
We fix a field K equipped with a discrete valuation val : K → Z ⊔ { + ∞} , normalized by val ( K × ) = Z . We assume that K iscomplete with respect to the distance defined by val. We let K ◦ bethe subring of K consisting of elements of nonnegative valuationand π be a uniformizer of K , that is an element of valuation 1. Weset k = K ◦ / πK ◦ . The Tate algebra K { X } is defined by: K { X } : = (cid:110) (cid:213) i ∈ N n a i X i s.t. a i ∈ K and val ( a i ) −−−−−−−→ | i |→ + ∞ + ∞ (cid:111) Series in K { X } have a natural analytic interpretation: they are an-alytic functions on the closed unit disc in K n . We recall that K { X } is equipped with the so-called Gauss valuation defined by:val (cid:16) (cid:213) i ∈ N n a i X i (cid:17) = min i ∈ N n val ( a i ) . Series with nonnegative valuation form a subring K { X } ◦ of K { X } .The reduction modulo π defines a surjective homomorphism ofrings K { X } ◦ → k [ X ] . Terms and monomials.
By definition, an integral
Tate term is anexpression of the form a X i with a ∈ K ◦ , a (cid:44) i ∈ N n . IntegralTate terms form a monoid, denoted by T { X } ◦ , which is abstractlyisomorphic to ( K ◦ \{ }) × N n . We say that two Tate terms a X i and b X j are equivalent when val ( a ) = val ( b ) and i = j . Tate termsmodulo equivalence define a quotient T { X } ◦ of T { X } ◦ , which isisomorphic to N × N n . The image in T { X } ◦ of a term t ∈ T { X } ◦ iscalled the monomial of t and is denoted by mon ( t ) .We fix a monomial order ≤ ω on N n and order T { X } ◦ ≃ N × N n lexicographically by block with respect to the reverse naturalordering on the first factor N and the order ≤ ω on N n . Pulling backthis order along the morphism mon, we obtain a preorder of T { X } ◦ that we shall continue to denote by ≤ . The leading term of a Tateseries f = (cid:205) a i X i ∈ K { X } ◦ is defined by: LT ( f ) = max i ∈ N n a i X i ∈ T { X } ◦ . We observe that the a i X i ’s are pairwise nonequivalent in T { X } ◦ ,showing that there is no ambiguity in the definition of LT ( f ) . The leading monomomial of f is by definition LM ( f ) = mon ( LT ( f )) . Gröbner bases.
The previous inputs allow us to define the notion ofGrobner basis for an ideal of K { X } ◦ . Definition 2.9.
Let I be an ideal of K { X } ◦ . A family ( д , . . . , д s ) ∈ I s is a Gröbner basis (in short, GB) of I if, for all f ∈ I , there exists i ∈ { , . . . , s } such that LM ( д i ) divides LM ( f ) . A classical argument shows that any GB of an ideal I generates I . The following theorem is proved in [CVV19, Theorem 2.19].Theorem 2.10. Every ideal of K { X } ◦ admits a GB. The explicit computation of such a GB is of course a centralquestion. It was addressed in [CVV19], in which the authors de-scribe a Buchberger algorithm and an F4 algorithm for this task.The aim of the present article is to improve on these results byintroducing signatures in this framework and eventually designF5-like algorithms for the computation of GB over Tate algebras.
Important remark.
For the simplicity of exposition, we chose torestrict ourselves to the Tate algebra K { X } and not consider thevariants K { X ; r } allowing for more general radii of convergence.However, using the techniques developed in [CVV19] (paragraph General log-radii of §3.2), all the results we will obtain in this articlecan be more generally extended to K { X ; r } . The goal of this section is to adapt the G2V algorithm to the set-ting of Tate algebras. Although all definitions, statements and al-gorithms are formally absolutely parallel to the classical setting,proofs in the framework of Tate algebras are more subtle, due tothe fact the orderings on Tate terms are not well-founded but onlytopologically well-founded. In order to accomodate this weakerproperty, we import ideas from [L+18] where the case of local ringsis considered.
We fix a monomial order ≤ ω of N n and write ≤ for the term or-der on T { X } ◦ it induces. We consider an ideal I in K { X } ◦ alongwith a GB G of I . Let f ∈ K { X } ◦ . We are interested in com-puting a GB of I = I + ⟨ f ⟩ . Mimicing what we have recalled in§2.1, we introduce the K { X } ◦ -sub-module M ⊂ K { X } ◦ × K { X } ◦ consisiting of pairs ( u , v ) such that u f − v ∈ I . The definitionsof regular reduction (Definition 2.1), strong Gröbner bases (Defi-nition 2.2), J-pair (Definition 2.4) and cover (Definition 2.5) extend verbatim to the context of Tate algebras, with the precaution thatthe leading monomial is now computed with respect to the order ≤ as explained in §2.2.Proposition 3.1. Suppose that G = {( u , v ) , . . . , ( u s , v s )} is anSGB of M . Then:(1) { u s.t. ( u , ) ∈ G } is a Gröbner basis of ( I : f ) . (2) { v s.t. ( u , v ) ∈ G for some u } is a Gröbner basis of I . Proof. Let G be an SGB of M.Let h ∈ ( I : f ) . Then h f ∈ I and ( h , ) ∈ M . By definition, since G is an SGB of M , there exists ( u , ) ∈ G such that LM ( u ) divides LM ( h ) . This implies the first statement of the proposition.Let now h ∈ I . If LM ( h ) ∈ I , there exists a pair ( , h ′ ) ∈ M with LM ( h ) = LM ( h ′ ) . This pair is divisible by some ( , v ) ∈ G ,proving that LM ( v ) divides LM ( h ′ ) = LM ( h ) in this case. We nowsuppose that LM ( h ) (cid:60) LM ( I ) . This assumption implies that any a ∈ K { X } ◦ with ( a , h ) ∈ M ( i.e. a f − h ∈ I ) must satisfy LM ( a ) ≥ LM ( h )/ LM ( f ) . We can then choose a series a ∈ K { X } ◦ such that ( a , h ) ∈ M and LM ( a ) is minimal for this property. Moreover, since G is an SGB, the pair ( a , h ) has to be top-reducible by some ( u , v ) ∈ lgorithm 1: G2V (resp. PoTe) algorithm input : f , . . . , f m in k [ X ] (resp. K { X } ◦ ) output: a GB of the ideal generated by the f i ’s Q ← ( f , . . . , f m ) GBasis ← ∅ for f ∈ Q do G ← {( , д ) : д ∈ GBasis } ∪ {( , f )} S ← { LM ( д ) : д ∈ GBasis } B ← { J-pair (( , f ) , ( , д )) : д ∈ GBasis } while B (cid:44) ∅ do pop ( u , v ) from B , with smallest u if ( u , v ) is covered by G then continue if u is divisible by some s ∈ S then continue v ← regular_reduce ( u , v , G ) if v = then add u to S else for ( s , д ) ∈ G do if J-pair (( u , v ) , ( s , д )) is defined then add J-pair (( u , v ) , ( s , д )) to B add ( u , v ) to G GBasis ← { v : ( u , v ) ∈ G } return GBasis G . If v (cid:44)
0, we deduce that LM ( v ) divides LM ( h ) . Otherwise, letting t = LT ( a )/ LT ( u ) , we obtain ( a − tu , h ) ∈ M with LM ( a − tu ) < LM ( a ) , contradicting the minimality of LM ( a ) . As a conclusion, wehave proved that LM ( v ) divides LM ( h ) in all cases, showing thatthe set { v s.t. ( u , v ) ∈ G for some u } is a GB of I . □ Theorem 3.2 (Cover Theorem).
Let G be a finite subset of M such that: • G contains ( , f ) ; • the set { д ∈ K { X } ◦ : ( , д ) ∈ G } forms a Gröbner basis of I .Then G is an SGB of M iff every J-pair of G is covered by G . The proof of Theorem 3.2 is presented in §3.2 below. Before this,let us observe that Theorem 3.2 readily shows that the G2V algo-rithm (see Algorithm 1, page 4) extends verbatim to Tate algebras.The resulting algorithm is called the PoTe algorithm. The correct-ness of the PoTe algorithm is clear thanks to Theorem 3.2. Its termi-nation is not a priori guaranteed because the call to regular_reduce may enter an infinite loop (see [CVV19, §3.1]). However, if we as-sume that all regular reductions terminate (which is guaranteed inpractice by working at finite precision), the PoTe algorithm termi-nates as well thanks to the Noetherianity of K { X } ◦ . Throughout this subsection, we consider a finite set G satisfyingthe assumptions of Theorem 3.2. PoTe means “ Po sition over Te rm”. Algorithm 2:
VaPoTe algorithm input : f , . . . , f m in K { X } ◦ output: a GB of the ideal generated by the f i ’s Q ← ( f , . . . , f m ) GBasis ← ∅ while Q (cid:44) ∅ do pop f from Q , with smallest valuation G ← {( , д ) : д ∈ GBasis } ∪ {( , f )} S ← { LM ( д ) : д ∈ GBasis } B ← { J-pair (( , f ) , ( , д )) : д ∈ GBasis } while B (cid:44) ∅ do pop ( u , v ) from B , with smallest u if ( u , v ) is covered by G then continue if u is divisible by some s ∈ S then continue v ← regular_reduce ( u , v , G ) if val ( v ) > val ( f ) then add u to S ; add v to Q else for ( s , д ) ∈ G do if J-pair (( u , v ) , ( s , д )) is defined then add J-pair (( u , v ) , ( s , д )) to B add ( u , v ) to G GBasis ← { v : ( u , v ) ∈ G } return GBasis
We first assume that G is an SGB of M . Let p , p ∈ G and write p i = ( u i , v i ) for i ∈ { , } . We set t = lcm ( LM ( v ) , LM ( v )) ∈ T { X } ◦ and t i = t / LM ( v i ) . If LM ( t u ) = LM ( t u ) , the J -pair of ( p , p ) is not defined and there is nothing to prove. Otherwise, if i (resp. j ) is the index for which LM ( t i u i ) is maximal (resp. LM ( t j u j ) is minimal), the J -pair of ( p , p ) is t i p i , which is regularly top-reducible by p j . Continuing to apply regular top-reductions by el-ements of G as long as possible, we reach a pair ( u , v ) ∈ M whichis no longer regularly top-reducible by any element of G and forwhich LM ( u ) = LM ( t i u i ) and LM ( v ) < LM ( t i v i ) . Since G isan SGB of M , ( u , v ) must be super top-reducible by some pair ( u , v ) ∈ G . By definition of super top-reducibility, LM ( u ) divides LM ( u ) = LM ( t i u i ) and LM ( v ) · LM ( u ) = LM ( v ) · LM ( u ) . Thisshows that LM ( v ) · LM ( u i ) < LM ( v i ) · LM ( u ) and then that ( u , v ) covers t i p i .We now focus on the converse and assume that each J -pair of G is covered by G . We define: W = (cid:8) ( u , v ) ∈ M , top-reducible by no pair of G (cid:9) and assume by contradiction that W is not empty.Lemma 3.3. The set W does not contain any pair of the form ( u , v ) with u = or LM ( v ) ∈ LM ( I ) . Proof. By our assumptions, if LM ( v ) ∈ LM ( I ) , v is reducibleby some д with ( , д ) ∈ G . In particular, ( u , v ) is top-reducible by ( , д ) and cannot be in W . If u =
0, then v ∈ I and we are reducedto the previous case. □ emma 3.4. Let p = ( u , v ) ∈ W . Then there exists a pair p = ( u , v ) ∈ G such that LT ( u ) divides LT ( u ) , say LT ( u ) = t LT ( u ) and t LT ( v ) is minimal for this property.Furthermore, t p is not regularly top-reducible by G . Proof. We have already noticed that u (cid:44)
0. Since ( , f ) ∈ G ,there exists a pair in G satisfying the first condition. Since G isfinite, there exists one that further satisfies the minimality condi-tion.We assume by contradiction that t p is regularly top-reducibleby G . Consider p = ( u , v ) ∈ G be a regular reducer of t p ,in particular there exists a term t such that t LT ( v ) = t LT ( v ) ,and t LT ( u ) < t LT ( u ) . The J-pair of p and p is then definedand equals to τ · ( u , v ) with τ dividing t . Write t = τt ′ forsome term t ′ . By hypothesis, this J-pair is covered, so there exists P = ( U , V ) ∈ G and a term θ such that θ · LT ( U ) = τ · LT ( u ) and θ · LT ( V ) < τ · LT ( v ) . As a consequence: t ′ θ · LT ( U ) = t · LT ( u ) = LT ( u ) t ′ θ · LT ( V ) < t · LT ( v ) . So t ′ P contradicts the minimality of p . □ Let ν be the minimal valuation of a series v for which ( u , v ) ∈ W .We make the following additional assumption: ν < + ∞ . In otherwords, we assume that W contains at least one element of the form ( u , v ) with v (cid:44)
0. We set: W = (cid:8) ( u , v ) ∈ W s.t. val ( LM ( v )) = ν (cid:9) . Lemma 3.5.
The set L = { LM ( u ) : ( u , v ) ∈ W } admits a minimalelement. Proof. We assume by contradiction that L does not have a min-imal element. Thus, we can construct a sequence ( u k , v k ) k ≥ withvalues in W such that LM ( u k ) is strictly decreasing. As a con-sequence, in the Tate topology, u k f converges to 0. Hence, for k large enough, val ( u k f ) > ν = val ( v k ) . From W ⊂ M , weget v k − u k f ∈ I and LM ( v k ) = LM ( v k − u k f ) ∈ LM ( I ) . ByLemma 3.3, this is a contradiction. □ Let W be the subset of W consisting of pairs ( u , v ) for which LM ( u ) is minimal. Note that by Lemma 3.3, this minimal value isnonzero.Lemma 3.6. For any ( u , v ) , ( u , v ) ∈ W , LM ( v ) = LM ( v ) . Proof. Let ( u , v ) and ( u , v ) in W , and assume that the lead-ing terms are not equivalent, that is LM ( v ) (cid:44) LM ( v ) . Withoutloss of generality, we can assume that LM ( v ) > LM ( v ) . By con-struction of W , LM ( u ) = LM ( u ) , that is LT ( u ) = aLT ( u ) forsome a ∈ K , val ( a ) =
0. Since u and u are nonzero, we can write u = LT ( u ) + r and u = LT ( u ) + r . Eliminating the leadingterms, we obtain a new element ( u ′ , v ′ ) = ( r − ar , v − av ) . By as-sumption, LM ( v ′ ) = LM ( v ) , and LM ( u ′ ) < LM ( u ) . Observe that ( u ′ , v ′ ) cannot be top-reduced by G as otherwise, ( u , v ) wouldalso be top-reducible by G . Hence ( u ′ , v ′ ) ∈ W , contradicting theminimality of LM ( u ) . □ Let now p = ( u , v ) ∈ W . From Lemma 3.4, there exists p = ( u , v ) ∈ G and a term t such that LT ( tu ) = LT ( u ) and tp is notregular top-reducible by G . We define p ∗ = ( u ∗ , v ∗ ) = p − tp = ( u , v ) − t ( u , v ) . We remark that LM ( u ∗ ) < LM ( u ) . Moreover LM ( v ) (cid:44) LM ( tv ) since otherwise p would be top-reducible by p , contradicting thefact that p ∈ W .We first examine the case where LM ( v ) < LM ( tv ) . It im-plies that LM ( v ∗ ) = LM ( tv ) > LM ( v ) . Let us prove first that p ∗ (cid:60) W . We argue by contradiction. From p ∗ ∈ W , we would de-rive val ( v ∗ ) ≥ ν = val ( v ) and then val ( v ∗ ) = val ( v ) since theinequality in the other direction holds by assumption. We con-clude by noticing that LM ( u ∗ ) < LM ( u ) contradicts the mini-mality of LM ( u ) . So p ∗ (cid:60) W , i.e. p ∗ is top-reducible by G . Let p = ( u , v ) ∈ G top-reducing p ∗ . If v =
0, then LM ( u ) divides LM ( u ∗ ) . Besides, the pair: p ′∗ = ( u ′∗ , v ∗ ) = (cid:0) u ∗ − LT ( u ∗ ) LT ( u ) u , v ∗ (cid:1) satisfies LM ( u ′∗ ) < LM ( u ∗ ) and thus cannot be in W either. Weiterate this process until we can only find a reductor q = ( U , V ) ∈ G with V (cid:44)
0. Let t = LM ( v ∗ )/ LM ( V ) . Then: t LM ( V ) = LM ( v ∗ ) = LM ( tv ) , t LM ( U ) ≤ LM ( u ∗ ) < LM ( tu ) if U (cid:44) q regularly top-reduces tp , which contradicts Lemma 3.4.Let us now move to the case where LM ( v ) > LM ( tv ) . Then LM ( v ∗ ) = LM ( v ) . Combining this with LM ( u ∗ ) < LM ( u ) , wededuce p ∗ (cid:60) W , i.e. p ∗ is top-reducible by G . As in the previouscase, we construct q = ( U , V ) ∈ G with V (cid:44) t suchthat: t LM ( V ) = LM ( v ∗ ) = LM ( v ) , t LM ( U ) ≤ LM ( u ∗ ) < LM ( u ) if U (cid:44) . Thus q regularly top-reduces p , which contradicts p ∈ W .As a conclusion, in both cases, we have reached a contradiction.This ensures that ν = + ∞ . In particulier, W contains an element p of the form ( u , ) . Let p = ( u , v ) ∈ G be given by Lemma 3.4.If v =
0, this pair would be a reducer of ( u , ) ∈ W , which is acontradiction. So v (cid:44)
0. Set t = LT ( u ) LT ( u ) . Let: p ∗ = ( u ∗ , v ∗ ) = ( u , ) − t ( u , v ) = ( u − tu , − v ) Then LM ( u ∗ ) < LM ( u ) and LM ( v ∗ ) = tLM ( v ) . From v (cid:44) p ∗ (cid:60) W . So p ∗ is top-reducible by p = ( u , v ) ∈ G ,meaning that there exists a term t such that t LM ( v ) = LM ( v ∗ ) = tLM ( v ) and t LM ( u ) ≤ LM ( u ∗ ) < tLM ( u ) . So p is a regular top-reducer of tp , which contradicts Lemma 3.4.Finally, we conclude that W is empty. By construction, G is anSGB of M . In this section, we design a variant of the PoTe algorithm in which,roughly speaking, signatures are first ordered by increasing valu-ations. .1 The VaPoTe algorithm
The VaPoTe algorithm is Algorithm 2. It is striking to observethat it looks formally very similar to the PoTe Algorithm (Algo-rithm 1) as they only differ on lines 3–4 and, more importantly,on lines 13–14. However, these slight changes may have signifi-cant consequences on the order in which the inputs are processed,implying possibly important differences in the behaviour of thealgorithms.The VaPoTe algorithm has a couple of interesting features. First,if we stop the execution of the algorithm at the moment whenwe first reach a series f of valuation greater than N on line 4,the value of GBasis is a GB of the image of I = ⟨ f , . . . , f m ⟩ in K { X } ◦ / π N K { X } ◦ . In other words, the VaPoTe algorithm can beused to compute GB of ideals of K { X } ◦ / π N K { X } ◦ (for our modi-fied order) as well.Secondly, Algorithm 2 remains correct if the reduction on line 12is interrupted as soon as the valuation rises. The property allowsfor delaying some reductions, which might be expensive at onetime but cheaper later (because more reductors are available). Italso has a theoretical interest because the reduction process may a priori hang forever (if we are working at infinite precision); in-terrupting it prematurely removes this defect and leads to moresatisfying termination results. We introduce some notations. For a series f ∈ K { X } ◦ , we write ν ( f ) = π − val ( f ) f (which has valuation 0 by construction) and de-fine ρ ( f ) as the image of ν ( f ) in K { X } ◦ / πK { X } ◦ ≃ k [ X ] . Moregenerally if A is a subset of K { X } ◦ , we define ν ( A ) and ρ ( A ) ac-cordingly.We consider f , . . . , f m ∈ K { X } ◦ and write I for the ideal gener-ated by f , . . . , f m . For any integer N , we set I N = I ∩( π N K { X } ◦ ) .Clearly I N + ⊂ I N for all N . Let ¯ I N be the image of π − N I N in k [ X ] ;we have a canonical isomorphism ¯ I N ≃ I N / I N + . Besides, the mor-phism I N → I N + , f (cid:55)→ π f induces an inclusion ¯ I N → ¯ I N + .Hence, the ¯ I N ’s form a nondecreasing sequence of ideals of k [ X ] .We define Q all as the set of all series that are popped from Q online 13 during the execution of Algorithm 2. Since the algorithmterminates when Q is empty, Q all is also the set of all series that hasbeen in Q at some moment. For an integer N , we further define: Q N = (cid:8) f ∈ Q all s.t. val ( f ) = N (cid:9) , Q ≤ N = (cid:8) f ∈ Q all s.t. val ( f ) ≤ N (cid:9) , Q > N = (cid:8) f ∈ Q all s.t. val ( f ) > N (cid:9) . Let also τ N be the first time we enter in the while loop on line 3with Q ⊂ π N K { X } ◦ . If this event never occurs, τ N is defined asthe time the algorithm exits the main while loop. We finally let GBasis N be the value of the variable GBasis at the checkpoint τ N .Lemma 4.1. Between the checkpoints τ N and τ N + :(1) the elements popped from Q are exactly those of Q N , and(2) the “reduction modulo π N + ” of the VaPoTe algorithm behaveslike the G2V algorithm, with input polynomials ρ ( Q N ) and initialvalue of GBasis set to ρ ( GBasis N ) . VaPoTe means “ Va luation over Po sition over Te rm” Proof. We observe that, after the time τ N , only elements withvaluation at least N + Q . The first statement then fol-lows from the fact that the elements of Q has popped by increasingvaluation. The second statement is a consequence of (1) togetherwith the fact that all f and v manipulated by Algorithm 2 betweenthe times τ N and τ N + have valuation N . □ Since the G2V algorithm terminates for polynomials over a field,Lemma 4.1 ensures that each checkpoint τ N is reached in finitetime if the call to regular_reduce does not hang forever. This latterproperty holds when we are working at finite precision and is alsoguaranteed if we interrupt the reduction as soon as the valuationraises.We are now going to relate the ideals ¯ I N with the sets Q N , Q ≤ N and Q > N . For this, we introduce the syzygies between theelements of ρ ( Q ≤ N ) . More precisely, we set: S N = (cid:110) ( a f ) f ∈ Q ≤ N s.t. (cid:213) f ∈ Q ≤ N a f ν ( f ) ≡ ( mod π ) (cid:111) . and let ¯ S N be the image of S N under the projection K { X } ◦ → k [ X ] ;in other words, ¯ S N is the module of syzygies of the set ρ ( Q ≤ N ) , i.e. ¯ S n = Syz ( ρ ( Q ≤ N )) with the notation of Definition 2.7. We alsodefine the linear mapping: φ N : ( K { X } ◦ ) Q ≤ N → K { X } ◦ ( a f ) f ∈ Q ≤ N (cid:55)→ (cid:213) f ∈ Q ≤ N a f ν ( f ) . By definition, φ N takes its values in the ideal generated by ν ( Q ≤ N ) and φ N ( S N ) ⊂ πK { X } ◦ .Proposition 4.2. For any integer N , the following holds:(a) The family ρ ( GBasis N + ) is a GB of ¯ I N .(b) φ N ( S N ) ⊂ (cid:10) π · ν ( Q ≤ N ) , π − N Q > N (cid:11) .(c) I N + = (cid:10) π N + · ν ( Q ≤ N + ) , Q > N + (cid:11) .(d) ¯ I N + = (cid:10) ρ ( Q ≤ N + ) (cid:11) . Proof. When N <
0, we have S N = I N + = I and ¯ I N = N and assume that the proposition holds for N −
1. By theinduction hypothesis, we know that ρ ( GBasis N ) is a GB of ¯ I N − .It then follows from Lemma 4.1 that ρ ( GBasis N + ) is a GB of theideal generated by ¯ I N − and ρ ( Q N ) , which is equal to ¯ I N by theinduction hypothesis. The assertion (a) is then proved.Between the checkpoints τ N and τ N + , each signature u addedto S on line 14 corresponds to a family ( a f ) f ∈ Q ≤ N for which thesum (cid:205) f a f f equals the element v added to Q on the same line.Rescaling the a f ’s, we cook up an element z ∈ S N with the prop-erty that φ N ( z ) = π − N v . Let Z ⊂ S N be the set of those ele-ments. From Proposition 2.3 and Lemma 2.8, we derive that ¯ S N isgenerated by ¯ S N − (viewed as a submodule of ¯ S N by filling newcoordinates with zeroes) and Z . Thus: φ N ( S N ) = φ N − ( S N − ) + (cid:10) φ N ( Z ) , π · ν ( Q ≤ N ) (cid:11) ⊂ φ N − ( S N − ) + (cid:10) π − N Q > N , π · ν ( Q ≤ N ) (cid:11) . The assertion (b) now follows from the induction hypothesis, oncewe have observed that Q > N − = π N ν ( Q N ) ∪ Q > N .et us now prove (c). Let h ∈ I N + . Then h ∈ I N and we can usethe induction hypothesis to write: h = π N (cid:213) f ∈ Q ≤ N a f ν ( f ) + (cid:213) д ∈ Q > N b д д for some a f , b д ∈ K { X } ◦ . Reducing modulo π N + , we find that thefamily ( a f ) f ∈ Q ≤ N belongs to S N . From (b), we deduce that: (cid:213) f ∈ Q ≤ N a f ν ( f ) ∈ (cid:10) π · ν ( Q ≤ N ) , π − N Q > N (cid:11) . Hence h ∈ (cid:10) π N + ν ( Q ≤ N ) , Q > N (cid:11) and we conclude by noticingthat Q > N = π N + ν ( Q N + ) ∪ Q > N + .Finally, (d) follows from (c) by dividing by π N + and reducingmodulo π . □ Termination.
Since k [ X ] is noetherian, the sequence of ideals ( ¯ I N ) is eventually constant. This implies that GBasis cannot grow indef-initely; in other words, the final value of
GBasis is reached in finitetime. However, the reader should be careful that this does not meanthat Algorithm 2 terminates. Indeed, once the final value of
GBasis has been computed, one still has to check that the remaining se-ries in Q reduce to zero; this is achieved by performing divisionsand can hang forever if we are working at infinite precision. Nev-ertheless, this misfeature seems very difficult to avoid since, whenworking at infinite precision, the input series contain themselvesan infinite number of coefficients and any modification on one ofthem could have a strong influence on the final result. Correctness.
Let G be the output of Algorithm 2, that is the limit ofthe ultimately constant sequence ( GBasis N ) . For a positive integer N , we define: G ≤ N = (cid:8) f ∈ G s.t. val ( f ) ≤ N (cid:9) . Since only elements of valuation at least N + GBasis after the checkpoint τ N + , we deduce that G ≤ N = GBasis N + .Hence, by Proposition 4.2, ρ ( G ≤ N ) is a GB of ¯ I N for all N ≥ G is in-deed a GB of I . For this, we consider f ∈ I . We write N = val ( f ) , sothat ρ ( f ) is the image in k [ X ] of π − N f . Moreover, we know that LM ( ρ ( f )) is divisible by LM ( ρ ( д )) for some д ∈ G ≤ N , i.e. there ex-ists i ∈ N n such that LM ( ρ ( f )) = X i · LM ( ρ ( д )) . This readily impliesthat: LM ( f ) = π N − val ( д ) · X i · LM ( д ) showing that LM ( д ) divides LM ( f ) in T { X } ◦ given that val ( д ) ≤ N .We have then proved that the leading monomial of any element of I is divisible by some LM ( д ) with д ∈ G , i.e. that G is a GB of I . We have implemented both the PoTe and VaPoTe algorithms inSageMath . Our implementation includes the following optimiza-tion: at the end of the loop ( i.e. after line 20), we minimize andreduce the current GB in construction. This operation is allowedsince all signatures are discarded after each iteration of the loop.Similarly, we reduce each new series f popped from Q on line 4before proceeding it. These ideas were explored in the algorithm https://trac.sagemath.org/ticket/28777 Parameters Buchberger PoTe VaPoTe p = ℓ =
5, prec =
12 87 . . . p = ℓ =
5, prec =
12 321 30 . . p = ℓ =
5, prec =
12 83 . . . p = ℓ =
7, prec = . . . p = ℓ =
7, prec = . . Table 1: Timings for the computation of GBs related to thetorsion of points of Tate curves (all times in seconds)
F5-C [EP10] and, as mentionned before, were one of the main mo-tivations for adopting an incremental point of view.Our implementation is also able to compute GB of ideals in K { X } . For this, we simply use a reduction (for no extra cost) tothe case of K { X } ◦ (see [CVV19, Proposition 2.23]). We also makemonic the signatures in S after each iteration of the main loop;in the PoTe algorithm, this renormalization gives a stronger covercriterion and thus improves the performances.As mentionned in Section 4.1, Algorithm 2 remains correct if thereductions are interrupted as soon as the valuation rises. This canbe done in the reduction step before processing the next f , beforeadding elements to the SGB, as well as in the inter-reduction step.Delaying reductions could be interesting, for instance, if the inputideal is saturated: indeed, in this case, the algorithm never con-siders elements with positive valuation and delayed reductions donot need to be done afterwards. On the other hand, performingmore reductions earlier leads to shorter reducers and potentiallyfaster reductions later. In practice, in our current implementation,we have observed all possible scenarios: interrupting the reduc-tions can make the computation faster, slower, or not make anysignificant difference. Numerous experimentations on various random inputs show thatthe VaPoTe algorithm performs slightly better than the PoTe algo-rithm on average. Besides, both PoTe and VaPoTe algorithms usu-ally perform much better than Buchberger algorithm, although weobserved important variations depending on the input system.As mentionned in the introduction, Tate algebras are the build-ing blocks of p -adic geometry. One can then cook up interestingsystems associated to meaningful geometrical situations. As a ba-sic example, let us look at torsion points on elliptic curves.We recall briefly that (a certain class of) elliptic curves over K = Q p are uniformized by the Tate curve (see [Ta95]), which can beseen as the curve defined over K { q } by the explicit equation y + xy = x + a x + a with: a = ∞ (cid:213) n = n ( pq ) n − ( pq ) n , a = ∞ (cid:213) n = n + n ( pq ) n − ( pq ) n . Given an auxiliary prime number ℓ , we consider the ℓ -th divisionpolynomial Φ ℓ ( x , q ) ∈ K { q } ◦ [ x ] associated to the Weierstrass formof the above equation. By definition, its roots are the abscissas of ℓ -torsion points of the Tate curve. We now fix p and ℓ and considerthe system in 3 variables Φ ℓ ( x , q ) = Φ ℓ ( x , q ) =
0. Its solutionsparametrize the pairs of elliptic curves sharing a common ℓ -torsionoint. Computing a GB of it then provides information about tor-sion points on p -adic elliptic curves.Table 1 shows the timings obtained for computing a GB of theabove systems for different values of p , ℓ and different precisions.We clearly see on these examples than both PoTe and VaPoTe over-perform the Buchberger algorithm. Faster reductions.
Observing how our algorithms behave, one im-mediately notices that reductions are very slow. It is not that sur-prising since our reduction algorithm is currently very naive. Forthis reason, we believe that several structural improvements arequite possible. An idea in this direction would be to store a well-chosen representative sample of reductions and reuse them lateron. Typically, we could cache the reductions of all terms of the form x e · · · x en n (with respect to the current GB in construction) anduse them to emulate a fast exponentation algorithm in the quotientring K { X } ◦ /⟨ GB ⟩ .Another attractive idea for accelerating reduction is to incor-porate Mora’s reduction algorithm [Mo82, MRW17] in our frame-work. Let us recall that Mora’s algorithm is a special method forreducing terms with respect to local or mixed orders ( i.e. ordersfor which there exist terms t < πr to our list of reductors eachtime we encounter a remainder r (including f itself) in the re-duction process. We believe that this optimization, if it is care-fully implemented, could already have some impact on the perfor-mances. Besides, observing that the equality f = r + πq f also reads f = ( + πq ) − r , we realize that Mora reduction of a Tate seriesare somehow related to its Weierstrass decomposition. Moreover,at least in the univariate case, it is well known that Weierstrassdecompositions can be efficiently computed using a well-suitedNewton iteration. It could be interesting to figure out whether thisstrategy extends to multivariate series and, more generally, to thecomputation of arbitrary Mora reductions. Using overconvergence properties.
In a different direction, we wouldlike to underline that the orderings we are working with are bydesign block orders (comparing first the valuation). However, inthe classical setting, we all know that graded orders often lead tomuch more efficient algorithms. Unfortunately, in the setting ofthis article, the very first definition of a Tate series already forcesus to give the priority to the valuation in the comparison of terms;otherwise, the leading term would not be defined in general.Nonetheless, we emphasize that if graded orders does not existover K { X } , they do exist over some subrings. Precisely, recall that,given a tuple r = ( r , . . . , r n ) , we have defined : K { X ; r } : = (cid:110) (cid:213) i ∈ N n a i X i s.t. a i ∈ K and val ( a i ) − r · i −−−−−−−→ | i |→ + ∞ + ∞ (cid:111) We refer to [CVV19] for more details where r · i denotes the scalar product of the vectors r and i . Whenthe r i ’s are all nonnegative, K { X ; r } embeds naturally into K { X } ;precisely, elements in K { X ; r } are those series that overconvergesover the polydisk of polyradius (| π | − r , . . . , | π | − r n ) . Moreover, thealgebra K { X ; r } is equipped with the valuation val r defined by:val r (cid:16) (cid:213) i ∈ N n a i X i (cid:17) = min i ∈ N n val ( a i ) − r · i . This valuation defines a new term ordering ≤ r . We observe that,from the point of view of K { X } , it really looks like a graded or-der: the quantity val r ( f ) plays the role of (the opposite of) a “totaldegree” which mixes the contribution of the valuation and that ofthe classical degree.In light of the above remarks, we formulate the following ques-tion. Suppose that we are given an ideal I ⊂ K { X } ◦ (say, of dimen-sion 0) generated by some series f , . . . , f m . If we have the promisethat the f i ’s all overconverge, i.e. all lie in K { X ; r } for a given r ,can we imagine an algorithm that computes a GB of I taking ad-vantage of the term ordering ≤ r ? As an extreme case, if we havethe promise that all the f i ’s are polynomials (that is r i = + ∞ forall i ), can one use this assumption to accelerate the computation ofa GB of I ? REFERENCES [BGR84] Bosch Siegfried, Günzter Ulrich and Remmert Reinhold, Non-Archimedeananalysis, Springer-Verlag (1984)[Bu65] Buchberger Bruno, Ein Algorithmus zum Auffinden der Basiselemente desRestklassenringes nach einem nulldimensionalen Polynomideal (An Algorithmfor Finding the Basis Elements in the Residue Class Ring Modulo a Zero Dimen-sional Polynomial Ideal), English translation in J. of Symbolic Computation,Special Issue on Logic, Mathematics, and Computer Science: Interactions. Vol.41, Number 3-4, Pages 475–511, 2006[CL14] Caruso Xavier and Lubicz David, Linear Algebra over Z p [[ u ]] and relatedrings, LMS J. Comput. Math. 17 (2014), 302-344[CVV19] Caruso Xavier, Vaccon Tristan and Verron Thibaut, Gröbner bases overTate algebras, in Proceedings: ISSAC’19.[EF17] Eder Christian and Faugère Jean-Charles, A survey on signature-based algo-rithms for computing Gröbner bases, J. of Symbolic Computation, 2017[EP10] Eder Christian and Perry John, F5C: A variant of Faugère’s F5 algorithm withreduced Gröbner bases, J. of Symbolic Computation, 2010[Fa99] Faugère Jean-Charles, A new efficient algorithm for computing Gröbner bases(F4), Journal of Pure and Applied Algebra, 1999[Fa02] Faugère, Jean-Charles, A new efficient algorithm for computing Gröbnerbases without reduction to zero (F5), in Proceedings: ISSAC’02.[GGV10] Gao Shuhong, Guan Yinhua and Volny IV Frank, A new incremental algo-rithm for computing Groebner bases, In Proceedings: ISSAC’10.[GVW16] Gao Shuhong, Volny IV Frank, and Wang Mingsheng, A new frameworkfor computing Gröbner bases, Mathematics of computation, 2016, vol. 85, no297, p. 449-465.[GR95] Gräbe Hans-Gert, Algorithms in Local Algebra, J. of Symbolic Computation , 1995, 545–557[L+18] Lu Dong, Wang Dingkang, Xiao Fanghiu, Zhou Jie Extending the GVW Algo-rithm to Local Ring, Proceedings of 43th International Symposium on Symbolicand Algebraic Computation, ISSAC’18, New York, USA[MRW17] Markwig Thomas, Ren Yue and Wienand Olivier, Standard bases in mixedpower series and polynomial rings over rings, J. of Symbolic Computation ,2017, 119–139[Mo82] Mora Ferdinando, An algorithm to compute the equations of tangent cones,Proceedings of European Computer Algebra Conference in Marseille, 1982, 158–165[NS01] Norton Graham H. and Sălăgean Ana, Strong Grobner bases and cyclic codesover a finite-chain ring, Electronic notes in discrete maths12