In complex geometry, the concept of analytic functions is obviously a notion of first importance. They form a class of functions that exhibit strong rigidity properties as polynomials do but, at the same time, allow for many analytic constructions such as taking limits, integrals, etc. For this reason, they often appear as a bridge between algebra and analysis.
For many arithmetical applications, the completion of is often as relevant as or . At the beginning of the 20th century, mathematicians realized that it would be quite interesting to develop the theory of -adic analytic functions and eventually that of -adic analytic geometry. However doing so is not an easy task owing to the unpleasant topology on , which is totally disconnected.
In (Tate, ), Tate proposed to replace the classical -adic topology by some well-suited Grothendieck topology and came up with the notion of -adic rigid variety. Basically, the construction of rigid varieties follows that of schemes in algebraic geometry. They are obtained by gluing pieces — the so-called affinoids — with respect to the aforementioned Grothendieck topology. As for affinoids, they are defined as the “spectrum” of quotients of some particular algebras, called Tate algebras. Thereby, Tate algebras play the same role in rigid geometry as polynomial algebras do in classical algebraic geometry.
From the purely algebraic point of view, Tate algebras have been widely studied and it has been demonstrated that they share some properties with polynomial algebras (BGR84, ). However, as far as we know, the computational aspects of Tate algebras have not been developed yet. This contrasts with the polynomial setting, for which we have at our disposal the theory of Gröbner bases (Bu65, ; Cox15, ), which has become over the years a research topic on its own. The aim of the present article is to extend the notion of Gröbner bases to Tate algebras.
Some difficulties need to be overcome. The most significant one is that elements in Tate algebras are, by nature, infinite convergent series and so they do not have a degree. This seems to be a serious obstruction since the degree is the most basic notion on which the classical theory of Gröbner bases is built. However, analyzing the definition of Tate algebras, we notice that a Tate series defines a sequence of polynomials (of growing degrees) by reduction modulo when varies. In order to take advantage of this observation, we introduce an order on the terms taking into account the -adic valuation of the coefficients. This order is not well-founded as classical term orders are usually. However, we shall prove that it is topologically well-founded (in the sense that every decreasing sequence tends to ) and that this weaker property is enough to guarantee the termination of our algorithms in the finite precision model.
Related works. Gröbner bases over rings — and in particular over and — have also received some attention (AL, ; KC, ). These developments are of course related to this article since quotients of Tate algebras are polynomial algebras over for varying. The main difference between our point of view and that of loc. cit. appears in the choice of the term ordering; while, in the theory of Gröbner bases of rings, only the degree is considered, our setting forces us to include the valuation of the coefficients in the definition of the term ordering. It is the “price to pay” to be able to pass smoothly to the completion and catch inexact bases as or .
The special term ordering we use comes from two different sources. The first one is the theory of tropical Gröbner bases by Chan and Maclagan (CM, ) in which, for the first time, the valuation of the coefficients has been taken into account in the definition of the term ordering. Later on, Vaccon and his coauthors (Vaccon:these, ; Vaccon:2015, ; Vaccon:2017, ; Vaccon:2018, ) observed that tropical orders are relevant for the computation of -adic Gröbner bases as they improve substantially the numerical accuracy. The definition of our term order is the natural outcome of this observation. Our second source of inspiration is the theory of standard bases, which was designed originally to “compute” the singularties of algebraic varieties (Mora, ; Grabe, ). This theory introduces the notion of term order of local/mixed type, on which the term ordering we are using in the present article is modeled.
Structure of the article. In §2, we introduce Tate algebras and develop the theory of Gröbner bases over them. We prove in particular the existence of finite Gröbner bases and study their structure. §3 is devoted to algorithms. We first design a variant of the Buchberger algorithm that runs over Tate algebras. Several results towards its numerical stability are also presented. We then move to F4-like algorithms and show how they could be adapted to fit into the framework of Tate algebras. Finally, in §4, an implementation in SageMath is briefly discussed.
Notations. The notation will refer to the set of nonnegative integers (including ). If is a ring, we will denote its group of invertible elements by . We fix a positive integer . Let be variables. We will use the short notation for . Similarly for , we shall write for .
2. Gröbner bases over Tate algebras
Throughout this article, we fix a field equipped with a discrete valuation , normalized by . We shall always 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 .
A typical example of as above is the field of -adic numbers (equipped with the -adic valuation). For this example, we have and .
2.1. Tate algebras
We endow with the usual scalar product.
Definition 2.1 ().
Let . The Tate algebra is defined by:
The tuple is called the convergence log-radii of the Tate algebra.
Elements of are the power series converging on the product of closed balls where is the absolute value on induced by . When , we will simply write instead of .
Example 2.2 ().
Let . The series lies in . The series does not lie in , because it does not converge when evaluated at (for example). However, it does converge when evaluated at with , so it lies in for all negative .
The Tate algebra is equipped with the Gauss valuation defined as follows:
We observe that the minimum is always reached thanks to the growth condition imposed in Definition 2.1. Moreover, the image of is discrete. Geometrically, the Gauss valuation corresponds to the minimal valuation reached by the series on its domain of convergence (possibly after a finite extension of ).
Definition 2.3 ().
The integral Tate algebra ring is defined as the subring of consisting of elements with nonnegative Gauss valuation.
Again we will use the notation for . When , observe that and similarly for . The case then reduces to via a change of variables.
Example 2.4 ().
With the notations of Example 2.2, does not lie in , but does lie in .
Proposition 2.5 ().
We have .
2.2. About terms
From now on, we fix a log-radii .
Monoids of terms.
We first recall some basic definitions.
Definition 2.6 ().
A monoid is a set equipped with a single associative binary operation, which has a neutral element.
An ideal of a monoid is a subset such that, for all and , we have .
We define the monoid of terms as the multiplicative monoid consisting of the elements with and . We let also be the submonoid of consisting of terms for which . The multiplicative group (resp. ) embeds into (resp. ). We set:
The inclusion induces a canonical morphism (which is no longer injective) . The ideals of (resp. of ) are in bijective correspondance with the ideals of (resp. of ). Moreover, and do not contain non trivial invertible elements. In other words, the divisibility relation defines an order on and . The following lemma elucidates the structure of and .
Lemma 2.7 ().
(1) The mapping , is an isomorphism of monoids.
(2) The mapping , is an injective morphism of monoids; its image is included in where is a common denominator of the coordinates of .
(3) The natural morphism corresponds to the projection onto the factor .
Proposition 2.8 ().
Let be an ideal of (resp. of ). Then there exists a unique subset of having the two following properties: (1) generates , and (2) every subset generating contains . Moreover is finite.
The unicity is easy. Indeed if and satisfy (1) and (2), one must have and , i.e. . In order to prove the existence, we define as the set of minimal elements of for the divisibility relation. The fact that generates follows from the fact that divisibility is a well-funded order on (cf Lemma 2.7). The point (2) is obvious.
It remains to prove that is finite. For this, we observe that any sequence with values in necessarily has a nondecreasing subsequence. Extracting subsequences repeatedly, we find that the previous property also holds for sequences with values in for any integer . By Lemma 2.7, it also holds for sequences with values in (resp. in ). Therefore, if were not finite, we would be able to extract from a nondecreasing sequence. This contradicts the fact that is composed by minimal elements. ∎
Definition 2.9 ().
Let be an ideal of (resp. of ). The subset of Proposition 2.8 is called the skeleton of ; it is denoted by .
The skeleton of an ideal of (resp. of ) is defined as the skeleton of its image in (resp. in ); it is denoted by .
In what follows, it will sometimes be convenient to work more generally with fractional ideals. By definition a fractional ideal of is a subset of which is stable by multiplication by elements in . The notion of skeleton can be extended to fractional ideals of for which there exists such that . For such ideals, is a finite subset of . An interesting example of fractional ideal is:
Remark 2.10 ().
The effective computation of is not an easy problem. It has been solved for in (CL, ) using the theory of continued fractions. It would be interesting to generalize the results of loc. cit. to higher .
We fix a monomial order on . We recall that this means that is a well-order which is compatible with the addition. Usual examples of monomial orders are lex, grevlex, etc.
Definition 2.11 ().
We define a preorder on , by:
Remark 2.12 ().
The inequality sign is reversed in the first line: we require that and not . This is not a typo and will be important in the sequel.
We underline that is not antisymmetric (and so not an order). More precisely, for , the fact that and is equivalent to the existence of such that . As a consequence, induces an order on . On the contrary, we draw the attention of the reader that does not factor through .
Example 2.13 ().
Let and consider with the lexicographical order. The preorder orders terms as follows:
The terms and are “equal” for . So are and .
It is easily seen that the preorder is total. In turns out that it is not a well-order since the infinite sequence is strictly decreasing. Nevertheless, we have:
Lemma 2.14 ().
Let be a strictly decreasing sequence in (resp. in ). Then .
From the definition of , it follows that the sequence is nondecreasing. Moreover it takes its values in for some positive integer . Finally, the fact that is a well-order implies that for each fixed , there is only a finite number of indices for which . Combining these inputs, we find that must tend to . ∎
We notice that if , the terms and are never “equal” for . Therefore, any nonzero series has a unique leading term. We denote it .
Example 2.15 ().
With the notations of Example 2.13, the leading term of is .
2.3. Gröbner bases
Definition 2.16 ().
Given an ideal of (resp. of ), we denote by the subset of (resp. of ) consisting of elements of the form with , .
We check immediately that is an ideal of the monoid (resp. of ).
Definition 2.17 ().
Let be an ideal of (resp. of ). A family is a Gröbner basis (in short, GB) of if is generated by the ’s in (resp. ).
Proposition 2.18 ().
Let be a GB of an ideal of (resp. of ). Then generates .
Let . We define inductively a sequence as follows. Let . Given , we write and define . Then . By Lemma 2.14, goes to infinity when goes to infinity. Therefore we can then write as a converging series. By regrouping terms, we get . ∎
Proposition 2.8 gives a lot of information about the ideal (where is an ideal of or ). These results have interesting consequences on Gröbner bases.
Theorem 2.19 ().
Any ideal of or has a finite GB.
Let be the elements of . For all , let be such that in (resp. in ). Then is a GB of . ∎
Remark 2.20 ().
Combining the previous theorem with Proposition 2.18, we obtain that any ideal of (resp. of ) is finitely generated. In other words, we have proved that the rings and are Noetherian (which was of course already known for a long time).
Another important consequence of Proposition 2.8 is the notion of minimal GB that we discuss now.
Definition 2.21 ().
Let be an ideal of (resp. of ). A GB is minimal if the images in (resp. in ) of the ’s are exactly the elements of , with no repetition.
A direct consequence of the definition is that two minimal GB of a given ideal have the same cardinality, namely the cardinality of . Proposition 2.8 also implies the next theorem.
Theorem 2.22 ().
Let be an ideal of (resp. of ). Let be a GB of . Then, there exists a subset which is a minimal GB of .
2.4. Comparison results
So far, we have defined a notion of GB for ideals of and . The aim of this subsection is to compare them.
Proposition 2.23 ().
Let be an ideal of and let be a GB of . Then is a GB of the ideal of .
Remark 2.24 ().
Note that minimality of GB is not preserved when passing from to . For example, is a minimal GB of the ideal of . However it is not a minimal GB of since divides in this ring.
Going in the other direction (i.e. from to ) is more subtle. First of all, we remark that, if we start with an ideal of , there exist many ideals of with the property that . However, the set of such ideals has a unique maximal element (for the inclusion); it is the ideal . This special ideal can also be caracterized by the fact that it is -saturated.
Proposition 2.25 ().
Let be an ideal of and let be a GB (resp. a minimal GB) of . We assume that for all . Then is a GB (resp. a minimal GB) of .
Let be a GB of . Let . Then is a multiple of one of the ’s in . Since , we deduce that divides in as well. Consequentlt is a GB of . The fact that minimality is preserved is easy. ∎
When , it is easy to build a GB of satisfying the assumption of Proposition 2.25 from any GB of . Indeed if is a GB of then is an integer for all and the family is a GB of . On the contrary, when , the problem is more complicated as illustrated by the next example.
Example 2.26 ().
Choose and and let be ideal of generated by . The ideal is then generated by and . More precisely, one checks that is a minimal GB of . In particular, we observe that the cardinality of a minimal GB of does not agree with that of a minimal GB of .
For a general , Proposition 2.25 can be refined as follows.
Proposition 2.27 ().
Let be an ideal of and let be a GB of . Then a GB of is ’s where, for each fixed , the ’s enumerate the elements of (cf Eq. (2)).
Reduction in the residue field.
When , the quotient is isomorphic to the polynomial algebra , on which we have a well-defined notion of Gröbner bases.
Proposition 2.28 ().
Let be an ideal of . Set and let be the image of in . Let in be such that and let be their images in . Then the following assertions are equivalent:
(1) is a GB of ;
(2) is a GB of ;
(3) is a GB of .
The equivalence between (1) and (2) has been already proved. We now prove that (2) implies (3). Let and let be a lift of . We can write for some and . Then . Therefore the ’s generate . We prove finally that (3) implies (2). Let . Set . Clearly and . Thus . By (3), we can write for and . We write with and similarly, with . Then divides . Let be such that . Then
with . This concludes the proof. ∎
3.1. Division and membership test
Not surprisingly, Gröbner bases can be used to test membership in ideals. Before going further in this direction, we need to adapt the division algorithm to our setting. We will need two variants depending on where we are looking for the quotients.
Proposition 3.1 ().
Let . Then, there exist (resp. ) and such that:
(2) for all and all terms of , in (resp. in ),
(3) for all terms of , we have .
We only give the proof of , the case of being totally similar. We will construct by induction sequences , () and such that:
We set , and . If is divisible by some , we set and , and leave unchanged and the others ’s. Otherwise, we set and .
In general, it does not terminate, keeping computing more and more accurate approximations of the ’s and . However, in the common case where the coefficients of the input series are all known up to finite precision, i.e. modulo for some , Algorithm 1 does terminate.
Remark 3.2 ().
When working at finite precision, it is more intelligent, instead of computing the quotient (which would possibly lead to losses of precision), to choose an exact term such that the equality holds at the working precision, and use it on lines 4 and 5. Doing so, we limit the losses of precision.
In general, the conditions of Proposition 3.1 are not enough to determine uniquely the ’s and . However, Proposition 3.3 below provides a weak unicity result when is a Gröbner bases, which can be used to test membership.
Proposition 3.3 ().
Let be an ideal of (resp. of ) and let be a GB of . Let . We assume that we are given a decomposition satisfying the requirements of Proposition 3.1. Then if and only if .
The “only if” is clear. Conversely, assume by contradiction that and . Then makes sense. From the conditions of Proposition 3.1, we deduce that is not divisible by for all . Hence . This is contradiction since . ∎
Remark 3.4 ().
In the integral Tate algebra setting, it is not true that the remainder in the division by Gröbner bases is unique. For example, the division in of by can be written either or . This is a general limitation of Gröbner bases over rings, even in the polynomial case (AL, ).
3.2. Buchberger’s algorithm
In this subsection, we adapt Buchberger’s algorithm to fit into the framework of Tate algebras. The adaptation is more or less straightforward except on two points. The first one is related to finite precision, as already encountered previously. The second point is of different nature; it is related to the fact that, when the log-radii are not integers, the crucial notion of S-polynomials is not well-defined as the monoid does not admit ’s. In what follows, we will give satisfying answers to these issues.
To begin with, we assume . Under this hypothesis, the monoid of terms admits ’s and ’s. Concretely we define:
where the and the over are taken coordinate by coordinate. In what follows, in order to simplify notations, we will write instead of . If and are two terms, the valuation of (resp. of ) is the minimum (resp. the maximum) of and .
Definition 3.5 ().
For in , we define:
We have the following classical lemma:
Lemma 3.6 ().
Let and . We assume that the ’s all have the same image in and that . Then
for some such that and for .
Theorem 3.7 ().
Let be elements of (resp. of ) and let be the ideal of (resp. of ) generated by the ’s. Then is a GB of if and only if all , , reduce to zero after division by using Algorithm 1.
The “only if” part follows from Proposition 3.3. We prove the “if” part. Let us assume by contradiction that there exists some such that . We can write with (resp. ). Define . We have because of the hypothesis that . We can moreover assume that the decomposition is chosen in such a way that is minimal.
Let be the set of indices for which for some . Set for and define ; we have . Applying Lemma 3.6, we find and terms , (for ) such that:
and , . Applying Proposition 3.1 with the S-polynomials, and using the fact that the leading terms of the summands in an S-polynomial cancel out, we get such that and for all . Therefore, we find that can be written as with and for all . This contradicts the minimality of . ∎
Studying its termination is a bit subtle. Indeed, we have already seen that Algorithm 1 does not terminate in general when we are working at infinite precision. Therefore, Algorithm 2 does not terminate either (since it calls Algorithm 1 on line 5). Nevertheless, one may observe that if, instead of calling Algorithm 1, we ask the reduced form of modulo to an oracle that answers instantly, then Algorithm 2 does terminate. In other terms, the only source of possible infinite loops in Algorithm 2 comes from Algorithm 1.
Of course, this point of view is purely theoretical and not satisfying in practice. In practice, the coefficients of are given at finite precision, i.e. modulo for some integer , and all the computations are carried out at finite precision. In this setting, we have seen that Algorithm 1 does terminate, so Algorithm 2 also terminates. The counterpart is that it is a priori not clear that the result output by Algorithm 2 is a correct approximation of a GB of the ideal we started with. Nevertheless, in the case of , this property holds true as precised by the following theorem.
Theorem 3.8 ().
Let be an ideal of and let be a generating family of . Let also be an integer such that