Universal Analytic Gröbner Bases and Tropical Geometry

A universal analytic Gröbner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gröbner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB’s for polynomial ideals, leaving open the question of how to compute them. In this paper, we provide an algorithm computing a UAGB for a given polynomial ideal, by traversing the Gröbner fan of the ideal. As an application, it offers a new point of view on algorithms for computing tropical varieties of homogeneous polynomial ideals, which typically rely on lifting the computations to an algebra of power series. Motivated by effective computations in tropical analytic geometry, we also examine local bases for more general convergence conditions, constraining the radii to a convex polyhedron. In this setting, we provide an algorithm to compute local Gröbner bases and discuss obstacles towards proving the existence of finite UAGBs.


INTRODUCTION
The notion of Tate algebras has been introduced by Tate in [25] to develop analytic geometry over the -adics, founding what is now called rigid geometry.This theory has proved to be central to many developments in number theory.In this context, Tate algebras and ideals in Tate algebras serve the same purpose as polynomial algebras and polynomial ideals in classical algebraic geometry.Tate algebras are defined as algebras of power series over a complete discrete valuation field with convergence conditions such as converging on a given ball or a polydisk with given radii.
In previous works [4][5][6], the authors showed that it is possible to define and compute Gröbner bases (GB) of ideals in Tate algebras, and modern algorithms for Gröbner bases computations like signature-based algorithms [5] and FGLM [6] can be adapted to this setting.
In [6,7], the authors paved the way for computations in Tate algebras in case of overconvergence, e.g.ideals defined by series converging on a bigger polydisk.Motivated by the application of analytic geometry in algebraic geometry, an extreme example of this phenomenon is that of ideals defined by polynomials in a Tate algebra.In [7], it was proved that it is possible to compute a GB of a polynomial ideal in a Tate algebra that is made of polynomials.It was also proved that for any polynomial ideal, there exists a universal analytic Gröbner basis (UAGB), i.e. a finite list of polynomials such that whenever they are seen as converging power series in a Tate algebra, they form a Gröbner basis of the corresponding ideal in this algebra.Such a UAGB of a polynomial ideal then contains abundant information on the local behavior of the ideal.In this paper, we prove that a UAGB is also universal irrespectively of the order used as tie-break in the algebra.Furthermore, we provide an algorithm to compute a UAGB in finite time (Algorithm 2, Theorem 4.7).
From a universal GB, it is natural to consider computing the tropical variety of an ideal.Over a field with valuation such as Q or F (( )), the tropical variety trop( ) of a variety defined by an ideal can be defined as the closure of the image of by the valuation, or alternatively using conditions on leading terms of the elements of .Acting as a combinatorial shadow of , many information on can be recovered from trop( ).The developments of tropical geometry have been plentiful.To only name a few: enumerative geometry [16], understanding optimization algorithms [1] or analyzing artificial neural networks of the ReLU type.
Universal GB can help in the computation of trop( ) using the second definition of the tropical variety.In our context, working with Tate algebras instead of polynomial rings gives rise to tropical analytic geometry.This emerging field has been defined in [21], adapting the language of tropical geometry to the world of rigid geometry.
In Section 5, we consider the case of tropical varieties of Tate polynomial ideals.We show that the tropical variety of a polynomial ideal is the union of the tropical varieties of its Tate completions, which allows to compute the tropical variety using universal analytical Gröbner bases and the Gröbner fan.This matches what was known for ideals in [[ ]] [X], which were used as a lifting target in existing algorithms for computing tropical varieties over valued fields.As such, we provide a new point of view on those algorithms, allowing them to work directly on the Tate series without lifting.To the best of our knowledge, this is the first effective application of tropical analytic geometry.
Finally, motivated by going further into the development of effective computations in rigid geometry, we aim at building up the tools for effective computations on affinoid subdomains.Roughly speaking, affinoid subdomains are constructed using generalizations of Tate algebras to more general convergence conditions (e.g.converging on an annulus) and taking quotients by ideals.
In Section 6, we make some steps into this journey by providing effective computations of local GB in the special case of some polyhedral subdomains as defined in [21].We conclude with some conjectures on UAGB in this context, along with examples and comments.

SETTING 2.1 Tate algebras and Gröbner bases
In this section, we recall the definition of Tate algebras and their theory of Gröbner bases (GB for short).Let be a field with a valuation val making it complete.Let be a uniformizer of , that is an element of valuation 1.Typical examples of such a setting are -adic fields like = Q with = or Laurent series fields like = Q(( )) with = .
For r = ( 1 , . . ., ) ∈ Q , the Tate algebra {X; r} is defined as We call the tuple r the convergence log-radii of the Tate algebra.We define the Gauss valuation of a term i X i as val r ( i X i ) = val( i )− r•i, and the Gauss valuation of i X i ∈ {X; r} as the minimum of the Gauss valuations of its terms.The valuation defines a metric on {X; r}, for which a sequence ( ) ∈N ∈ {X; r} converges to zero iff val r ( ) − −−−−− → →+∞ +∞.
In this article, we shall frequently need to consider all terms with minimal valuation together.
and the initial part of is By definition, for ∈ {X; r}, in r ( ) is a polynomial.We fix a classical monomial order ≤ on the set of monomials X i , which will be used for tie-breaks.Given two terms X i and X j (with , ∈ × ), we write X i < r, X j if val r ( X i ) > val r ( X j ), or val r ( X i ) = val r ( X j ) and X i < X j .By definition, the leading term of a Tate series = i X i ∈ {X; r} is its maximal term, and is denoted by LT r, ( ).Its coefficient and its monomial are denoted LC r, ( ) and LM r, ( ), with LT r, ( ) = LC r, ( ) × LM r, ( ).For , ∈ {X; r}, we define their S-polynomial as S-Poly( , ) = LT r, ( ) ( , ) − LT r, ( ) ( , ) where ( , ) = gcd(LT r, ( ), LT r, ( )).
A Gröbner basis (or GB for short) of an ideal of {X; r} is a set ⊆ such that for all ∈ , there exists an index ∈ such that LT r, ( ) divides LT r, ( ).A finite Gröbner basis ( 1 , . . ., ) is reduced if all LT r, ( )'s are monic, minimally generate LT r, ( ) and, for any , LT r, ( ) is the only term of in LT r, ( ).
The following theorem was proved in [4].
Theorem 2.2.Let be an ideal of {X; r}, then admits a finite Gröbner basis.

Local Gröbner bases of polynomial ideals in Tate algebras
For a polynomial ideal ⊂ [X] and a system of convergence log-radii r, we define r to be the ideal of {X; r} generated by the polynomials of .It is the completion of with respect to val r .The ideal r usually contains many series and polynomials not in .However, as is dense in r , it was proved in [7] that r does not contain more leading terms than , and that r admits a Gröbner basis comprised of polynomials.
Definition 2.3.Let ⊂ [X] be an ideal, r a system of log-radii and r the completion of in {X; r}.An r-local Gröbner basis of is a Gröbner basis of r comprised only of polynomials.
If one needs to vary the convergence log-radii, the following object is of interest: Definition 2.4.Let ⊂ [X] be an ideal.A finite set ⊂ ⊂ [X] such that for any r ∈ Q , is an r-local GB of is called a universal analytic Gröbner basis of (UAGB for short).
In the usual setting, it is required that universal Gröbner bases be a Gröbner basis for all monomial orders.Here, the definition requires only that all convergence radii be covered, without any restriction on the tie-breaking monomial order.However, we prove in Lemma 3.4 that given a polynomial ideal, any term ordering can be achieved with a suitable choice of a system of convergence radii.Finally [7] culminated with the following result: Theorem 2.5.Let ⊂ [X] be an ideal.Then there exists a universal analytic Gröbner basis of .
While the proof was not constructive, we provide in this article an algorithm to compute a UAGB of any polynomial ideal.

Homogenization and dehomogenization
Our algorithm to compute a UAGB of a polynomial ideal will rely on homogenization and dehomogenization.We consign here notations and basic properties taken from [7, §3.3] Definition 2.6.Let (•) * and (•) * be the homogenization and dehomogenization applications between [X] and [X, ].If ⊂ [X] is an ideal, we define * ⊂ [X, ] to be the ideal spanned by the * for ∈ .
Given r ∈ Q and ≤ a monomial order, we extend the term order < r, to [X, ] and {X, ; r, 0} as follows.
Definition 2.7.Given two terms X and , we write that X < (r,0), if one of the following holds: ) and X < X .This defines a term order on {X, ; r, 0}.

TERM ORDERS 3.1 Convergence radii and term orders
In this section, we collect different results regarding term orders in Tate algebras.First, we consider the relation between term orders and systems of convergence log-radii, and show that given finite data (e.g. a finite set of polynomials or an ideal), it is always possible to realize a term order by a suitable choice of system of convergence log-radii.
. Given a term order <, we define LT < ( ) = {LT < ( ) : ∈ }.We say that two term orders < 1 and < 2 on [X] are equivalent with respect to if Let ⊆ [X] be an ideal, we say that two term orders < 1 and < 2 on [X] are equivalent with respect to if The two relations are connected as follows.

P .
Let < 1 and < 2 be two term orders equivalent w.r.t. .Since is a GB of w.r.t.< 1 and < The following lemma states that modulo equivalence w.r.t., it is always possible to choose a term order determined only by the convergence condition.Lemma 3.3.Let r ∈ Q and let ≤ r, be a term order defined by val r and a tie-breaking order ≤ .Let be a finite set of terms in In particular, if is a finite set in [X], any equivalence class of term orders w.r.t.contains a term order < s, such that for all ∈ , LT s, ( ) = in s ( ).

P
. Thanks to [20, Th. 1] (see also [12,Lem. 1.3.1]),there exists some u ∈ Q such that for all 1 X ≠ 2 X in , Let s = r − u.Then, for any 1 = 1 , 2 = 2 X in such that 1 > r, 2 , one of the following is true: . Finally, if = and val( 1 ) = val( 2 ), then 1 X ≯ 2 X and there is nothing to prove.Therefore, s satisfies the claim.
The consequence for equivalence classes w.r.t.follows, by setting = ∈ Supp( ).Lemma 3.4.Let ⊆ [X] be an ideal, < r, 1 be a term order and a local Gröbner basis of w.r.t.< r, 1 .Then there exists s ∈ Q such that, for any tie-break order < 2 : • is a Gröbner basis of w.r.t.< s, 2 ; Let be the set of all terms which appear in and in the course of Buchberger's algorithm with weak normal form with as input w.r.t. the < r, 1 ordering.By Lemma 3.3, there exists s such that all terms in have distinct s-valuation, compatible with the order < r, 1 .Let < 2 be a tie-break order.
Note that when running Buchberger's algorithm with WNF, the radii of convergence are only used for determining leading terms.So if we run the algorithm with as input and for the order < s, 2 , all comparisons will be the same, the exact same terms will appear, and the result will be the same: all the S-polynomials have a weak normal form of 0 w.r.t., and so is a GB of w.r.t.< s, 2 .
By Lemma 3.2, this, together with the fact that the elements of have the same leading term for both orders, implies that the term orders are equivalent w.r.t. .
For such an s, we say that s defines a term order for .The tiebreaking order < 2 becomes irrelevant, and we omit it from the notations.In the rest of the paper, unless specified otherwise, we will always be considering orders < s where s defines a term order.
We then define N ( ) to be the Minkowski sum of the N ( )'s.
Remark 3.6.For = = 1, this coincides with the classical definition of the Newton polygon (up to a symmetry).Lemma 3.7.For the convex polyhedron N ( ), ∈ N ( ) is a vertex if and only if there is some = (1, 1 , . . ., ) ∈ Q +1 such that • is the unique minimum of the • 's for ∈ N ( ).
Proposition 3.8.The vertices of N ( ) are in one-to-one correspondence with the equivalence classes of term orders with respect to .

P
. For any ∈ 1, , we write = =1 , X , .Let us define the following set of index vectors: For j ∈ J, and ∈ 1, , we define , = 1, \ { } and we define , so that for any r ∈ j and ∈ 1, , LT r ( ) = , X ,j .Then, thanks to Lemma 3.3, there is one equivalence class of term orders with respect to for each non empty j .
This is enough to conclude that there is a one-to-one correspondance between vertices of N ( ) and equivalence classes of term orders with respect to .

UNIVERSAL ANALYTIC GRÖBNER BASES 1 4.1 Testing whether a set is a UAGB
The results of Section 3.2 are enough to immediately provide us with a procedure for deciding whether a set is a UAGB (Algorithm 1).
if and only if for any r in the equivalence classes of term orders with respect to , is a GB of r .In particular, Algorithm 1 is correct.

Computing a UAGB
We now show how to use that procedure to compute a UAGB.To that end, we recall the following result from [7].
Theorem 4.2.[7, Thm 7.6] Let ⊂ [X] be an ideal.Then the set Terms( for ∈ {vertices of } do if ∃ ∈ u not reducible modulo for the order < u then 6: return (False, u) ; 7: return True ; The proof of the previous theorem relies on the following lemma, which we also need.

P
. Let ∈ .Since is reduced and Unlike in the classical case, it is not in general possible to guarantee that any polynomial ideal admits a reduced local Gröbner basis for any convergence radii.However, for homogeneous ideals, [7, Lem.7.2] guarantees that there is a reduced local GB comprised only of polynomials, which can then be computed using any GB algorithm from [4,5,7,8,[26][27][28].

P
. Firstly, due to being a dehomogenization of homogeneous elements of ℎ , the ℎ , * 's are in (it is enough to dehomogenize an homogeneous combination of the * ).
Secondly, by [7,Cor 5.4], it is enough to check that for any ∈ , LT r ( ) is divisible by one of the LT r (ℎ , * )'s.
We can now provide an algorithm for computing UAGBs.r := system of log radii such that is not a GB of r (as produced by TestUAGB); := ∪ ; 6: return { * for ∈ } ; Theorem 4.7.Algorithm 2 is correct and computes a UAGB in finite time.Furthermore, if the input polynomials are homogeneous, the UAGB contains a reduced Gröbner basis for all orders.
Let us assume that TestUAGB( ) fails because is not an r-local GB of for some r.Then, thanks to Theorem 4.2 and Lemma 4.3, LT( ≤ ) ≠ LT( r ) for any integer , 1 ≤ ≤ and LT( ≤ ) = LT( r ) for some integer , < ≤ .Up to renumbering, we may assume that = + 1.Let be the reduced GB of for r.Then ∪ contains a reduced GB for the orders ≤ 1 , . . ., ≤ +1 .
We then prove by induction that, after at most calls to ReducedGB and to TestUAGB, the algorithm outputs such that contains a reduced GB of for each of ≤ 1 , . . ., ≤ and hence, is a UAGB of .
Finally, thanks to Lemma 4.5, the dehomogenization of is a UAGB of = * .If the input polynomials are homogeneous, the homogenization and dehomogenization steps are trivial, and the property that the UAGB contains a reduced GB for all orders is preserved.
Remark 4.8.From the proof of Theorem 4.7, Algorithm 2 needs at most #Terms( ) loops to compute a UAGB.Each loop may however cause many GB computations as it is unclear how the edges of the Newton polytopes vary along the computation.

Examples
one finds that ℎ is not a GB for the weight [0, 2, 0].The corresponding reduced GB will add the polynomial 2 − and ℎ = ] is enough to be a UAGB.

TROPICAL GEOMETRY 5.1 Analytical tropical varieties
In this section, we show that tropical geometry on Tate polynomial ideals specializes that of classical polynomial ideals.In particular, the results of Section 3 give us a Tate analogue of the Gröbner fan, and allow us to generalize the results of [17,22] for computing tropical varieties in [[ ]] [X], to any Tate algebra.First, we recall the classical notions of tropical geometry, and state their natural generalization to Tate algebras.Definition 5.1.Let w = ( 0 , . . ., ) ∈ R <0 × R be a system of weights.For a monomial = 1  1 • • • and ∈ , we define its weighted degree For ∈ [X], let deg w ( ) = max(deg w ( ) : ∈ Supp( )).We define the initial form of as Let ⊂ [X] be an ideal.Then in w ( ), the initial ideal of with respect to a system of weights w, is the ideal spanned by all in w ( ) for in .The tropical variety associated to is then defined as trop( ) = {w ∈ R <0 × R : in w ( ) does not contain a monomial}.
We say that the system of weights w and the system of log-radii r are compatible if r = − 1 0 , . . ., 0 .Conversely, given a system of log-radii r, the system of weights (−1, 1 , . . ., ) is compatible with r.The definitions above extend naturally to series and ideals in {X; r} by restricting to systems of weights which are compatible with r.
Remark 5.2.In particular, trop( r ) is either empty or a half-line formed of all the systems of weights compatible with r.
If the systems of weights w and the system of log-radii r are compatible, then for any term , This implies that deg w is a graduation: for any terms , ′ , deg w ( + The main result of this section is the fact that the tropical variety associated to is the union of the tropical varieties of all its completions r .Lemma 5.3.Let w be a system of weights, let r be the compatible system of convergence radii and let ≤ be a monomial order.Let ∈ {X; r}.Then: (1) in w ( ) = in r ( ), and in particular it is a polynomial; (2) LT ≤ (in w ( )) = LT r,≤ ( );

P
. By compatibility between the system of weights and the convergence log-radii, val r ( X i ) = −1 0 deg w ( X i ), and val r ( ) = −1 0 deg w ( ).So in w ( ) is the sum of all terms with minimal Gauss valuation in the support of , which is by definition in r ( ).The rest follows from the convergence properties in Tate series and the definition of the term order.Theorem 5.4.Let ⊆ [X] be an ideal.Let w ∈ R <0 × R be a system of weights, and let r = −( 1 / 0 , . . ., / 0 ) ∈ R be the compatible system of convergence log-radii.Let r ⊆ {X; r} be the completion of .Then and in particular, w ∈ trop( ) if and only if w ∈ trop( r ).Globally, trop( ) = s∈R trop( s ).
There exist series 1 , . . ., such that = 1 1 + • • • + .Let = val r ( ), and write each series as ℎ + , where ℎ is the sum of all terms with Gauss valuation at most − val r ( ) and = − ℎ has Gauss valuation greater than − val r ( ).By the convergence property, the ℎ 's are polynomials.The decomposition of becomes where the latter group consists exclusively of terms with Gauss valuation greater than .So none of those terms can appear in the initial form of , and as a consequence, in w ( ) = in w (ℎ , there exists series 1 , . . ., ∈ r and series 1 , . . ., such that From the above, we know that in w ( ) ∈ in w ( ) for all .Since ℎ is a polynomial, it has a maximal Gauss valuation .Similarly to before, any term in with Gauss valuation greater than −val r ( ) cannot appear in ℎ, so those terms must add to zero on the right hand side, and we can assume that the cofactors are polynomials, and therefore ℎ ∈ in w ( ) and the second inclusion is proved.
The rest of the statement follows by definition.

Analytic Gröbner fan
Similarly to the polynomial case, tropical varieties can be computed using the Gröbner fan of the ideal.In this section, we recall those definitions.The relation between tropical varieties and Gröbner fans is the same as in the usual case, namely, that initial forms generalize leading terms.Definition 5.5.Let w ∈ R <0 × R be a system of weights.Let ≤ be a term order on [X].We say that ≤ refines w if for all terms 1 , 2 , deg w ( 1 ) ≥ deg w ( 2 ) =⇒ 1 ≥ 2 .We say that w defines a term order for a finite set of polynomials or an ideal if r = −( 1 / 0 , . . ., / 0 ) defines a term order for that set or ideal.
If w defines a term order for a finite set , for all ∈ and all term orders ≤ refining w, LT ≤ ( ) = in w ( ).Similarly, if w defines a term order for an ideal , then for all term orders ≤ refining w, LT ≤ ( ) = in w ( ).
As seen in Lemma 3.3, for any finite set of polynomials or any ideal , and for any monomial order, there exists an equivalent monomial order defined by a system of weights.Definition 5.6.Let ⊂ [X] be an ideal.Let w be a system of weights.The analytic Gröbner cone w ( ) associated to w and is the set of all systems of weights w ′ such that in w ( ) = in w ′ ( ).The analytic Gröbner fan of is the fan given by all the analytic Gröbner cones of .Proposition 5.7.If w ∈ trop( ), then w ( ) ⊂ trop( ).In particular, the tropical variety associated to is a subfan of the Gröbner fan of .

P
. Whether a system of weights lies in the tropical variety only depends on the initial forms, and therefore applies identically to the cone.
Similar to the classical case, it allows to compute the tropical variety associated to by traversing the Gröbner fan.The following properties are transpositions of corresponding facts in the classical setting, and describe the Gröbner fan.
Proposition 5.8.Let ⊂ [X] be an ideal.Let w be a system of weights, r the convergence radii associated to w, and ≤ a term ordering refining w.Let be a reduced r-local Gröbner basis of (with ≤ as tie-break).Then: (1) in w ( ) = in w ( ) : ∈ ; (2) for any system of weights w 1 , w 1 ∈ w ( ) iff for all ∈ , in w ( ) = in w 1 ( ); (3) w ( ) is the relative interior of a polyhedral convex cone; (4) the closure of w ( ) in R <0 × R is the union of all cones (5) if w 1 and w 2 are two systems of weights such that w 1 ∩ w 2 {0} × R and w 1 ≠ w 2 , then there exists w 3 such that w 1 ∩ w 2 = w 3 , and it is a facet of both cones; (6) w ( ) has maximal dimension + 1 if and only if w defines a monomial order.

P
. For (1), let ∈ .Let r be the canonical image of in {X; r}, it lies in r .Since is an r-local GB of , its embedding r in r is a Gröbner basis of r w.r.t.≤.In particular, r reduces to 0 modulo r , which in particular implies that there exists a finite sequence of reductions with r, ∈ r for all , such that val r ( r, ) > val r ( r ) and val r ( r, ) = val r ( r ) for < .In particular, in w ( r ) = =1 in w ( r, ).This is a polynomial equality, which translates into in w ( ) = =1 in w ( ).
From there, the proof is similar to that of [22,Prop. 3.1.17].Let ∈ .Because is reduced, LT ≤ 1 ( ) = LT ≤ ( ) and in particular this term lies in both the support of in w ( ) and in w 1 ( ).Consider in w ( ) − in w 1 ( ), it lies in in w 1 ( ) and its leading term w.r.t.≤ 1 cannot be LT ≤ 1 ( ).Again since is reduced, this implies that in w ( ) − in w 1 ( ) = 0.
Finally, for (6), if w defines a monomial order for , by considering the finite set Supp( ), there exists a small enough ∈ R >0 such that for all u ∈ R <0 × R , and for all 1 , 2 ∈ Supp( ), deg w . Therefore w ( ) contains a ball of radius , so it is open and it must have maximal dimension.Conversely, if w does not define a monomial order, there exists a ∈ such that in w ( ) has at least two terms 1 and Since we know that has finitely many sets of leading terms by Th. 4.2, the analytical Gröbner fan of has only finitely many cones of maximal dimension, and therefore it is finite.If furthermore is homogeneous, then admits a universal Gröbner basis containing a reduced Gröbner basis for all orders, computable using Algorithm 2. In that case, by Prop.3.8, the vertices of N ( ) are in one-to-one correspondence with the equivalence classes of term orders with respect to , and by Lemma 3.2, this allows to compute all the equivalence classes of term orders with respect to .From there one can compute the cones of maximal dimension in the analytic Gröbner fan of , and finally all cones in the fan.Then, for each cone in the fan, one can pick a system of weights w in .From the property (1), in w ( ) is generated by the initial parts of elements of .Finally, we can decide whether in w ( ) contains a monomial by computing a basis of (in w ( ) : ( 1 • • ) ∞ ), using the algorithms in [7].
Just like in the classical case, both with and without valuation, this algorithm is not the most efficient, because the Gröbner fan can be significantly larger than the tropical variety.In [22] and [17], better algorithms have been presented for ideals in [[ ]] [X], and generalized to -adic fields by lifting back to that case.Those algorithms still traverse the Gröbner fan, but not in an exhaustive way, and rely on Buchberger's algorithm with Mora reductions for computing bases in the cones.
The conclusion of this section is that the key properties of the Gröbner fan in [[ ]] [X] are shared across polynomial rings over a valued ring or field, by way of Tate completions, and Buchberger's algorithm with weak normal form allows to compute generators in a similar way.Thus, it offers an alternative point of view on existing algorithms using liftings to reduce the problem to [[ ]] [X], instead performing the computations directly in the desired ring or field.
We expect that the other optimized algorithms for computing tropical varieties in [[ ]] [X] similarly transpose to the general setting.
Remark 5.9.The requirement that the ideal is homogeneous can be relaxed into requiring that the ideal admits a reduced Gröbner basis for all orders.This requirement is also present in the aforementioned works in [[ ]] [X].

TATE ALGEBRAS ON POLYHEDRAL SUBDOMAINS
As of now, we have studied Tate algebras using convergence conditions of the following types: (1) on a polydisk defined by log-radii s ∈ Q ; (2) convergence everywhere, i.e. the algebra [X]; (3) on all polydisks r ≤ s (overconvergence).
To build up the tools for rigid geometry or tropical analytic geometry as in [21], we need more general convergence conditions such as convergence on an annulus (e.g.converging for all X ∈ Q such that ∀ , ≤ val( ) ≤ for some , ∈ R) or converging on a polyhedron.
Following [21,Definition 6.3] but restricting to the case of power series instead of Laurent series, we define the following ring of functions {X; } for the rational points of a polyhedron.This corresponds to a special case of the affinoid algebra defined by a polyhedral subdomain in [21].The series in {X; } are exactly the series converging on all polydisks with radius given by the weights which are points of .We give an example in Figure 1.Proposition 6.2.If P is the convex hull of the 1 +R <0 , . . ., +R <0 then {X; } = =1 {X; s }.

P
. The ⊂ inclusion is clear.For the converse inclusion, we remark that if r ∈ , there are some ≥ 0 such that =1 = 1 and =1 s ≥ r.Consequently, for any term X ∈ [X], =1 val s ( X ) ≤ val r ( X ).

Local Gröbner bases
In this section, we explain how if P is such a polyhedron, = P ∩ Q , an ideal in {X; }, and r ∈ , it is possible to compute an r-local Gröbner basis of comprised only of elements of {X; }.First, we adapt the notion of écarts from [7, §6.1].

Figure 1 :
Figure 1: An example of a polyhedron with P the convex hull of the + R 2 <0 's in R 2 .