Signature-based algorithms for Gröbner bases over Tate algebras

02/11/2020 ∙ by Xavier Caruso, et al. ∙ Johannes Kepler University Linz Université de Limoges 0

Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gröbner bases over Tate algebras has been introduced and effectively implemented. One of the bottleneck in the algorithms was the time spent on reduction , which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gröbner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidences.



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1. Introduction

For several decades, many computational questions arising from geometry and arithmetics have received much attention, leading to the development of more and more efficient algorithms and softwares. A typical example is the development of the theory of Gröbner basis, which provides nowadays quite efficient tools for manipulating ideals in polynomial algebras and, eventually, algebraic varieties and schemes. At the intersection of geometry and number theory, one finds -adic geometry and, more precisely, the notion of -adic analytic varieties first defined by Tate in 1971 (Tate, ), which plays quite an important role in many modern theories and achievements (e.g. -adic cohomologies, -adic modular forms).

The main algebraic objects upon which Tate’s geometry is built are Tate algebras and their ideals. In an earlier paper (CVV, ), the authors started to study computational aspects related to Tate algebras: they introduce Gröbner bases in this context and design two algorithms (adapted from Buchberger’s algorithm and the F4 algorithm, respectively) for computing them.

In the classical setting, the main complexity bottleneck in Gröbner basis computations is the time spent reducing elements modulo the basis. The most costly reductions are typically reductions to , because they require successively eliminating all terms from the polynomial; yet their output has little value for the rest of the algorithm. Fortunately, it turns out that many such reductions can be predicted in advance (for example those coming from the obvious equality ) by keeping track of some information on the module representation of elements of an ideal, called their signature. This idea was first presented in Algorithm F5 (F5, ) and led to the development of many algorithms showing different ways to define signatures, to use them or to compute them. The interested reader can look at (EF17, ) for an extensive survey.

The Tate setting is not an exception to the wisdom that reductions are expensive. The situation is actually even worse since reductions to are theorically the result of an infinite sequence of reduction steps converging to . In practice, the process actually stops because we are working at finite precision; however, the higher the precision is, the more expensive the reductions to  are, for no benefit. This observation motivates investigating the possibility of adding signatures to Gröbner basis algorithms for Tate series.

Our contribution. In this paper, we present two signature-based algorithms 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 adopts an incremental point of view and uses the so-called cover criterion (GVW16, ) to detect reductions to . A key difficulty in the Tate setting is that the usual way to handle signatures assumes the constant term is the smallest one. However, the assumption fails in the Tate setting. We solve this issue by importing ideas from (LWXZ, ) in which the case of local algebras is addressed.

In the classical setting, incremental algorithms have the disadvantage of sometimes computing larger Gröbner bases for intermediate ideals, only to discard them later on. In order to mitigate this misfeature, the F5 algorithm uses a signature ordering taking into account the degree of the polynomials first, in order to process lower-degree elements first. In the Tate setting, the degree no longer makes sense and a better measure of progression of the algorithms is the valuation. Nonetheless, similarly to the classical setting, an incremental algorithm could perform intermediate computations to high valuation and just discard them later on. The second algorithm we will present, called the VaPoTe (valuation over position over term) algorithm, uses an analogous idea to that of F5 to mitigate this problem.

Organization of the article. In §2, we recall the basic definitions and properties of Tate algebras and Gröbner basis over them, together with the principles of the G2V algorithm. The two next sections are devoted to the PoTe and the VaPoTe algorithms respectively: they are presented and their correctness and termination are proved. Finally, implementation, benchmarks and possible future improvements, are discussed in §5.

Notations. Throughout this article, we fix a positive integer and use the short notation for . Given , we shall write for .

2. Ingredients

In this section, we present the two main ingredients we are going to mix together later on. They are, first, the G2V (GGV10, ) and GVW (GVW16, ) signature-based algorithms, and, second, the Tate algebras and the theory of Gröbner bases over them as developed in (CVV, ).

2.1. The G2V algorithm

In what follows, we present the G2V algorithm which was designed by Gao, Guan and Volny IV in (GGV10, ) as an incremental variant of the classical F5 algorithm. Our presentation includes the cover criterion which was formulated later on in (GVW16, ) by Gao, Volny IV and Wang. The incremental point of view is needed for the application we will discuss in §4. Moreover we believe that it has two extra advantages: first, it leads to simplified notations and, more importantly, it shows clearly where intermediate inter-reductions are possible.

Let be a field and denote the ring of polynomials over with indeterminates . We endow with a fixed monomial order . Let be an ideal in . Let be a Gröbner basis of with respect to . Let . We aim at computing a GB of the ideal Let be the -sub-module defined by the such that . The leading monomial of is the signature of .

Definition 2.1 (Regular reduction).

Let and be in . We say that is top-reducible by if

  1. either and divides ,

  2. or , divides and:

The corresponding top-reduction is

where is the first case and in the second case. This top-reduction is called regular when , that is when the signature of the reduced pair agrees with that of ; it is called super otherwise.

Definition 2.2 (Strong Gröbner bases).

A finite subset of is called a strong Gröbner basis (SGB, for short) of if any nonzero is top-reducible by some element of .

The G2V strategy derives the computation of a Gröbner basis through the computation of an SGB. They are related through the following proposition.

Proposition 2.3 ().

Suppose that is an SGB of Then:

  1. is a Gröbner basis of

  2. is a Gröbner basis of

To compute an SGB, we rely on J-pairs instead of S-polynomials.

Definition 2.4 (J-pair).

Let and be two elements in such that . Let and set for . Then:

  if , the J-pair of is ,

  if , the J-pair of is ,

  if , the J-pair of is not defined.

Definition 2.5 (Cover).

We say that is covered by if there is a pair such that divides and:

Theorem 2.6 (Cover Theorem).

Let be a finite subset of such that:

  • contains ;

  • the set forms a Gröbner basis of .

Then is an SGB of iff every J-pair of is covered by .

This theorem leads naturally to the G2V algorithm (see (GGV10, , Fig. 1)) which is rephased hereafter in Algorithm 1 (page 1). We underline that, in Algorithm 1, the SGB does not entirely appear. Indeed, we remark that one can always work with pairs in place of , reducing then drastically the memory occupation and the complexity. The algorithm maintains two lists and which are related to the SGB in construction as follows: is equal to the set of all when runs over the SGB. The criterion coming from the cover theorem is implemented on lines 2 and 2: the first (resp. the second) statement checks if is covered by an element of (resp. an element of ).

Syzygies. The G2V algorithm does not give direct access to the module of syzygies of the ideal. However, it does give access to a GB of (see Proposition 2.3), from which one can recover partial information about the syzygies, as shown below.

Definition 2.7 ().

Given , we define

Lemma 2.8 ().

Let generating and let generating . For , we write

and define . Then


Let . Then and we can write Then the syzygy has its last coordinate equal to and thus belongs to , which is enough to conclude. ∎

2.2. Tate algebras

Definitions. We fix a field equipped with a discrete valuation , normalized by . We assume that is complete with respect to the distance defined by . We let be the subring of consisting of elements of nonnegative valuation and be a uniformizer of , that is an element of valuation . We set . The Tate algebra is defined by:


Series in have a natural analytic interpretation: they are analytic functions on the closed unit disc in . We recall that is equipped with the so-called Gauss valuation defined by:

Series with nonnegative valuation form a subring of . The reduction modulo defines a surjective homomorphism of rings .

Terms and monomials. By definition, an integral Tate term is an expression of the form with , and . Integral Tate terms form a monoid, denoted by , which is abstractly isomorphic to . We say that two Tate terms and are equivalent when and . Tate terms modulo equivalence define a quotient of , which is isomorphic to . The image in of a term is called the monomial of and is denoted by .

We fix a monomial order on and order lexicographically by block with respect to the reverse natural ordering on the first factor and the order on . Pulling back this order along the morphism , we obtain a preorder of that we shall continue to denote by . The leading term of a Tate series is defined by:

We observe that the ’s are pairwise nonequivalent in , showing that there is no ambiguity in the definition of . The leading monomomial of is by definition .

Gröbner bases. The previous inputs allow us to define the notion of Grobner basis for an ideal of .

Definition 2.9 ().

Let be an ideal of . A family is a Gröbner basis (in short, GB) of if, for all , there exists such that divides .

A classical argument shows that any GB of an ideal generates . The following theorem is proved in (CVV, , Theorem 2.19).

Theorem 2.10 ().

Every ideal of admits a GB.

The explicit computation of such a GB is of course a central question. It was addressed in (CVV, ), in which the authors describe a Buchberger algorithm and an F4 algorithm for this task. The aim of the present article is to improve on these results by introducing signatures in this framework and eventually design F5-like algorithms for the computation of GB over Tate algebras.

Important remark. For the simplicity of exposition, we chose to restrict ourselves to the Tate algebra and not consider the variants allowing for more general radii of convergence. However, using the techniques developed in (CVV, ) (paragraph General log-radii of §3.2), all the results we will obtain in this article can be more generally extended to .

3. Position over term

The goal of this section is to adapt the G2V algorithm to the setting of Tate algebras. Although all definitions, statements and algorithms are formally absolutely parallel to the classical setting, proofs in the framework of Tate algebras are more subtle, due to the fact the orderings on Tate terms are not well-founded but only topologically well-founded. In order to accomodate this weaker property, we import ideas from (LWXZ, ) where the case of local rings is considered.

3.1. The PoTe algorithm

We fix a monomial order of and write for the term order on it induces. We consider an ideal in along with a GB of . Let . We are interested in computing a GB of . Mimicing what we have recalled in §2.1, we introduce the -sub-module consisiting of pairs such that . The definitions of regular reduction (Definition 2.1), strong Gröbner bases (Definition 2.2), J-pair (Definition 2.4) and cover (Definition 2.5) extend verbatim to the context of Tate algebras, with the precaution that the leading monomial is now computed with respect to the order as explained in §2.2.

Proposition 3.1 ().

Suppose that is an SGB of Then:

  1. is a Gröbner basis of

  2. is a Gröbner basis of


Let be an SGB of M.

Let Then and . By definition, since is an SGB of , there exists such that divides . This implies the first statement of the proposition.

Let now . If , there exists a pair with . This pair is divisible by some , proving that divides in this case. We now suppose that . This assumption implies that any with (i.e. ) must satisfy . We can then choose a series such that and is minimal for this property. Moreover, since is an SGB, the pair has to be top-reducible by some . If , we deduce that divides . Otherwise, letting , we obtain with , contradicting the minimality of . As a conclusion, we have proved that divides in all cases, showing that the set is a GB of . ∎

Theorem 3.2 (Cover Theorem).

Let be a finite subset of such that:

  • contains ;

  • the set forms a Gröbner basis of .

Then is an SGB of iff every J-pair of is covered by .

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 algorithm (see Algorithm 1, page 1) extends verbatim to Tate algebras. The resulting algorithm is called the PoTe111PoTe means “Position over Term”. algorithm. The correctness of the PoTe algorithm is clear thanks to Theorem 3.2. Its termination is not a priori guaranteed because the call to regular_reduce may enter an infinite loop (see (CVV, , §3.1)). However, if we assume that all regular reductions terminate (which is guaranteed in practice by working at finite precision), the PoTe algorithm terminates as well thanks to the Noetherianity of .

input :  in (resp. )
output : a GB of the ideal generated by the ’s
1   for  do
2       while  do
3             pop from , with smallest if  is covered by  then continue if  is divisible by some  then continue if  then
4                   add to
5            else
6                   for  do
7                        if  is defined then
8                              add to
10                  add to
return GBasis
Algorithm 1 G2V (resp. PoTe) algorithm
input :  in
output : a GB of the ideal generated by the ’s
1 while  do
2       pop from , with smallest valuation while  do
3             pop from , with smallest if  is covered by  then continue if  is divisible by some  then continue if  then
4                   add to ;  add to
5            else
6                   for  do
7                        if  is defined then
8                              add to
10                  add to
return GBasis
Algorithm 2 VaPoTe algorithm

3.2. Proof of the cover theorem

Throughout this subsection, we consider a finite set satisfying the assumptions of Theorem 3.2.

We first assume that is an SGB of . Let and write for . We set and . If , the -pair of is not defined and there is nothing to prove. Otherwise, if (resp. ) is the index for which is maximal (resp. is minimal), the -pair of is , which is regularly top-reducible by . Continuing to apply regular top-reductions by elements of as long as possible, we reach a pair which is no longer regularly top-reducible by any element of and for which and . Since is an SGB of , must be super top-reducible by some pair . By definition of super top-reducibility, divides and . This shows that and then that covers .

We now focus on the converse and assume that each -pair of is covered by . We define:

and assume by contradiction that is not empty.

Lemma 3.3 ().

The set does not contain any pair of the form with or .


By our assumptions, if , is reducible by some with . In particular, is top-reducible by and cannot be in . If , then and we are reduced to the previous case. ∎

Lemma 3.4 ().

Let . Then there exists a pair such that divides , say and is minimal for this property.

Furthermore, is not regularly top-reducible by .


We have already noticed that . Since , there exists a pair in satisfying the first condition. Since is finite, there exists one that further satisfies the minimality condition.

We assume by contradiction that is regularly top-reducible by . Consider be a regular reducer of , in particular there exists a term such that , and . The J-pair of and is then defined and equals to with dividing . Write for some term . By hypothesis, this J-pair is covered, so there exists and a term such that and . As a consequence:

So contradicts the minimality of . ∎

Let be the minimal valuation of a series for which . We make the following additional assumption: . In other words, we assume that contains at least one element of the form with . We set:

Lemma 3.5 ().

The set admits a minimal element.


We assume by contradiction that does not have a minimal element. Thus, we can construct a sequence with values in such that is strictly decreasing. As a consequence, in the Tate topology, converges to . Hence, for large enough, . From , we get and . By Lemma 3.3, this is a contradiction. ∎

Let be the subset of consisting of pairs for which is minimal. Note that by Lemma 3.3, this minimal value is nonzero.

Lemma 3.6 ().

For any , .


Let and in , and assume that the leading terms are not equivalent, that is . Without loss of generality, we can assume that . By construction of , , that is for some , . Since and are nonzero, we can write and . Eliminating the leading terms, we obtain a new element . By assumption, , and . Observe that cannot be top-reduced by as otherwise, would also be top-reducible by . Hence , contradicting the minimality of . ∎

Let now From Lemma 3.4, there exists and a term such that and is not regular top-reducible by . We define

We remark that . Moreover since otherwise would be top-reducible by , contradicting the fact that .

We first examine the case where . It implies that . Let us prove first that . We argue by contradiction. From , we would derive and then since the inequality in the other direction holds by assumption. We conclude by noticing that contradicts the minimality of . So , i.e. is top-reducible by . Let top-reducing . If , then divides . Besides, the pair:

satisfies and thus cannot be in either. We iterate this process until we can only find a reductor with . Let . Then:

Thus regularly top-reduces , which contradicts Lemma 3.4.

Let us now move to the case where . Then . Combining this with , we deduce , i.e. is top-reducible by . As in the previous case, we construct with and a term such that:

Thus regularly top-reduces , which contradicts .

As a conclusion, in both cases, we have reached a contradiction. This ensures that . In particulier, contains an element of the form . Let be given by Lemma 3.4. If , this pair would be a reducer of , which is a contradiction. So . Set . Let:

Then and . From , we deduce . So is top-reducible by , meaning that there exists a term such that and . So is a regular top-reducer of , which contradicts Lemma 3.4.

Finally, we conclude that is empty. By construction, is an SGB of .

4. Valuation over position over term

In this section, we design a variant of the PoTe algorithm in which, roughly speaking, signatures are first ordered by increasing valuations.

4.1. The VaPoTe algorithm

The VaPoTe222VaPoTe means “Valuation over Position over Term” algorithm is Algorithm 2. It is striking to observe that it looks formally very similar to the PoTe Algorithm (Algorithm 1) as they only differ on lines 12 and, more importantly, on lines 22. However, these slight changes may have significant consequences on the order in which the inputs are processed, implying possibly important differences in the behaviour of the algorithms.

The VaPoTe algorithm has a couple of interesting features. First, if we stop the execution of the algorithm at the moment when we first reach a series

of valuation greater than on line 4, the value of GBasis is a GB of the image of in . In other words, the VaPoTe algorithm can be used to compute GB of ideals of (for our modified order) as well.

Secondly, Algorithm 2 remains correct if the reduction on line 2 is interrupted as soon as the valuation rises. The property allows for delaying some reductions, which might be expensive at one time but cheaper later (because more reductors are available). It also has a theoretical interest because the reduction process may a priori hang forever (if we are working at infinite precision); interrupting it prematurely removes this defect and leads to more satisfying termination results.

4.2. Proof of correctness and termination

We introduce some notations. For a series , we write (which has valuation by construction) and define as the image of in . More generally if is a subset of , we define and accordingly.

We consider and write for the ideal generated by . For any integer , we set . Clearly for all . Let be the image of in ; we have a canonical isomorphism . Besides, the morphism , induces an inclusion . Hence, the ’s form a nondecreasing sequence of ideals of .

We define as the set of all series that are popped from on line 2 during the execution of Algorithm 2. Since the algorithm terminates when is empty, is also the set of all series that has been in at some moment. For an integer , we further define:

Let also be the first time we enter in the while loop on line 2 with . If this event never occurs, is defined as the time the algorithm exits the main while loop. We finally let be the value of the variable GBasis at the checkpoint .

Lemma 4.1 ().

Between the checkpoints and :

(1) the elements popped from are exactly those of , and

(2) the “reduction modulo ” of the VaPoTe algorithm behaves like the G2V algorithm, with input polynomials and initial value of GBasis set to .


We observe that, after the time , only elements with valuation at least are added to . The first statement then follows from the fact that the elements of has popped by increasing valuation. The second statement is a consequence of (1) together with the fact that all and manipulated by Algorithm 2 between the times and have valuation . ∎

Since the G2V algorithm terminates for polynomials over a field, Lemma 4.1 ensures that each checkpoint is reached in finite time if the call to regular_reduce does not hang forever. This latter property holds when we are working at finite precision and is also guaranteed if we interrupt the reduction as soon as the valuation raises.

We are now going to relate the ideals with the sets , and . For this, we introduce the syzygies between the elements of . More precisely, we set:

and let be the image of under the projection ; in other words, is the module of syzygies of the set , i.e. with the notation of Definition 2.7. We also define the linear mapping:

By definition, takes its values in the ideal generated by and .

Proposition 4.2 ().

For any integer , the following holds:

(a) The family is a GB of