Computing tropical varieties over fields with valuation

We show how tropical varieties of ideals I over a field K with non-trivial valuation can be traced back to tropical varieties of ideals in R[[t]][x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that generators are polynomial in t. While doing so, we also comment on the computation of the Groebner polytope structure and p-adic Groebner bases using our framework. All algorithms solely rely on existing standard basis techniques.


Introduction
Tropical varieties are commonly described as combinatorial shadows of their algebraic counterparts, and computing tropical varieties is an algorithmically highly challenging task, requiring sophisticated techniques from computer algebra and convex geometry. The first techniques were developed by Bogart, Jensen, Speyer, Surmfels and Thomas [BJS + 07], who focused on homogeneous ideals over C with the trivial valuation, which allowed them to rely on classical Gröbner basis methods. Furthermore, the authors showed that, under sensible conditions, their techniques can be used over the field of Puiseux series C{{t}} with its natural valuation, by regarding t as a variable in the polynomial ring instead of a uniformizing parameter in the coefficient ring. The inhomogeneity of the resulting ideal in C[t, x] can be worked around through homogenization and dehomogenization. In order to adapt these techniques to the field of p-adic numbers and the p-adic valuation, Chan and Maclagan adapted the classical theory of Gröbner bases [ChM13] to take the valuation on the ground field into account, instead of solely relying on monomial orderings. In this article, in Section 2, we discuss another approach to compute tropical varieties over an arbitrary field with valuation, which can be regarded as a generalisation of the trick used for C{{t}}. For that, we combine the existing notions of tropical varieties over power series [Tou05,BaT07,PPS13] with the concept of tropical varieties over coefficient rings [MaS15,Section 1.6]. Compared to [ChM13], the approach relies on existing standard basis theory, which not only allows us to exploit the highly optimized implementations that exist in many established computer algebra systems such as Singular [DGPS16] or Macaulay2 [GrS16], it also gives us access to a highly active field of research. Moreover, in Section 3, we improve on the techniques in [BJS + 07] by avoiding homogenization and dehomogenization. We also touch upon the topic on how to compute p-adic Gröbner bases in our framework. In Section 4 we present the algorithms for computing tropical varieties and in Section 5 we touch upon possible optimizations that are exclusive to non-trivial valuations. All algorithms in this article are implemented in the Singular library tropical.lib [JMMR16], relying on the gfanlib interface gfan.lib [Jen11,JRS16] for computations in convex geometry. They are publicly available as part of the official Singular distribution.

Tracing tropical varieties to a trivial valuation
The aim of this section is to show how tropical varieties over valued fields can be traced back to tropical varieties over integral power series. The linchpin of the section is to show that initial ideals over valued fields can be described through initial ideals over integral power series, the remaining results then follow naturally from this. To fix the notation, we will begin by recalling some very basic notions in tropical geometry that are of immediate relevance to us.

Convention 2.1
For the remainder of the article, fix a complete field K with non-trivial discrete valuation ν : K → R ∪ {∞} and a uniformizing parameter p ∈ K. Let O K be its ring of integers and let K denote its residue field. Let R ≤ O K be a dense, noetherian subring. By Cohen's Structure Theorem, we have two exact sequences Moreover, fix a multivariate polynomial ring K[x] = K[x 1 , . . . , x n ]. By abuse of notation, we will also use π to refer to both the map R t

Example 2.2 (p-adic numbers)
The most important example is the field K := Q p of p-adic numbers with O K := Z p the ring of p-adic integers. Then R := Z ≤ Z p is a natural dense subring, which is computationally easy to work over. The exact sequences in Convention 2.1 merely reflect the presentation of p-adic integers as power series in p: (1) K = k((t)) the field of Laurent series over a field k with O K = k t the ring of power series over k, R = k[t] and p = t; e.g. k = F q with q a prime power, as used in [SpS04,Section 7] or [Kal13], or k = Q as considered in [BJS + 07], see Example 2.14.
(2) Finite extensions K of Q p and F q ((t)), i.e. all local fields with non-trivial valuation, and also all higher dimensional local fields. (3) O K any completion of a localization of a Dedekind domain R at a prime ideal P R, p ∈ P a suitable element. Note that p does not need to generate P and hence O K need not be the completion with respect to the ideal generated by p, e.g. R = Z[ √ −5], P = 2, 1 + √ −5 and p = 2. (4) For an odd choice of R, consider K := Q(s)((t)) so that is multiplicatively closed as the complement of two prime ideals. Then R is a non-catenarian, dense subring of O K .
Definition 2.4 (initial forms, initial ideals, tropical varieties over valued fields) For a polynomial f = α∈N n c α · x α ∈ K[x] and a weight vector w ∈ R n , we define the initial form of f with respect to w to be: For any subset I ⊆ K[x] and a weight vector w ∈ R n , we define the initial ideal of I with respect to w to be: We refer to the set of weight vectors for which the initial ideal contains no monomial as the tropical variety of I, is the support of a pure polyhedral complex of same dimension that is connected in codimension 1.
Next, we will introduce tropical varieties in R t [x], and show how a certain class of them relates to tropical varieties in K[x]. In particular, we will note that those tropical varieties in R t [x] are pure and connected in codimension 1. We begin by introducing initial forms and initial ideals in R t [x] and show how they can be used to describe their pendants in K[x].
Definition 2.6 (initial forms, initial ideals) and a weight vector w ∈ R <0 × R n , we define the initial form of f with respect to w to be Given an ideal I R t [x] and a weight vector w ∈ R <0 × R n , we define the initial ideal of I with respect to w to be: This can be thought of as a natural extension of Definition 2.4 with trivial valuation on the coefficients. Note that we only allow weight vectors with negative weight in t, so that our result lies in a polynomial ring.
Example 2.7 (p-adic numbers) Let us consider the example in [Cha13, Chapter 3.6], the ideal over the 3-adic number Q 3 , so that and the weight vector (−1, w) ∈ R <0 × R 4 , w := (1, 11, 3, 19). A short computation yields in (−1,w) (π −1 I) = 3, x 2 1 , tx 1 x 3 x 4 , t 3 x 1 x 2 2 x 3 , t 4 x 1 x 4 2 , t 3 x 4 3 x 2 4 , and the similarity to the initial ideal of I under the 3-adic valuation is no coincidence: Proposition 2.8 For any ideal I O K [x] and any weight vector w ∈ R n , we have: where (·) denotes the canonical projection (·) : And because the valued weighted degree in O K [x] and the weighted degree in simply by applying the above argument to each of its terms. ⊆: Once again consider a term s = β c β p β · x α ∈ O K [x] with p ∤ c β for all β ∈ N. Then any preimage of it under π is of the form s ′ = β c β t β x α + r for some where β 0 := min{β ∈ N | c β = 0}. Now suppose deg (−1,w) (r) = deg (−1,w) ( β c β t β x α ). First observe that because t is weighted negatively, there can be no cancellation amongst the highest weighted terms of r and the terms of β c β t β x α , as the terms of β c β t β x α are not divisible by p, unlike the terms of the highest weighted terms of r. Therefore, we have Either way, we always have in (−1,w) (s ′ )| t=1 ∈ in ν,w (s) for any arbitrary preimage s ′ ∈ π −1 (s), and, as before, the same hence holds true for any arbitrary element

Corollary 2.9
For any ideal I K[x] and any weight vector w ∈ R n , we have: where (·) denotes the closure in the euclidean topology.

Example 2.11
Unlike over coefficient fields, initial ideals over coefficient rings may be devoid of monomials t β x α , β ∈ N and α ∈ N n while containing terms c · t β x α , c / ∈ R * . Consequently, tropical varieties over rings need not be pure. Consider the principal ideal generated by g := x + y + 2z ∈ Z t [x, y, z]. Figure 1 shows the intersection of its tropical variety with an affine subspace of codimension 2. Because g is homogeneous in x, y, z, its tropical variety is invariant under translation by (0, 1, 1, 1), and since t does not occur in g, it is also closed under translation by (−1, 0, 0, 0). Hence, the remaining points are then uniquely determined up to symmetry. Since in (−1,−1,−1,0) (g) = 2z is no monomial, the entire lower left quadrant is included in our tropical variety, while the two other maximal cones are not. However, because in (−1,1,1,0) (g) = x + y is no monomial either, the edge containing it is also part of our tropical variety. Therefore, the tropical variety cannot be the support of a pure polyhedral complex. Note, however, that I is not the type of ideal we are interested in, i.e. the type of ideal occurring in the following theorem.
Theorem 2.12 Let I K[x] be an ideal. The projection R ≤0 × R n → R n induces a natural bijection Proof. For the bijection, we show that in ν,w (I) monomial free ⇐⇒ in (−1,w) (π −1 I) monomial free.
Hence tx α + (t − 1)t · g 2 − t · r 2 , t 2 · r 2 ∈ in (−1,w) (π −1 I) and, as above, Thus r l , x α + (t − 1) · g 1 + t · r l ∈ in (−1,w) (π −1 I) and, more importantly, shaving off the the r l layer this time. Continuing this pattern eventually yields In this case we can use a combination of the steps in the previous cases to see In either case, we see that in (−1,w) (π −1 I) contains a monomial.

Corollary 2.13
If I K[x] defines an irreducible subvariety of (K * ) n of dimension d, then T (π −1 I) is the support of a pure polyhedral fan of dimension d + 1 connected in codimension one.
Example 2.14 Let K := Q((u)) be the field of Laurent series, equipped with is natural valuation ν u , and let I K[x, y] be the principal ideal generated by (x + y + 1) · (u 2 x + y + u).
Then T νu (I) is the union of two tropical lines, one with vertex at (0, 0) and one with vertex at (1, −1).
Hence T (π −1 I) is as shown in Figure 2, the cone over T νu (I). The polyhedral complex consists of 6 rays and 8 two-dimensional cones in a way that the intersection with the affine hyperplane yields a highlighted polyhedral complex, T νu (I).
, whose preimage is given by The tropical variety of the preimage is combinatorially of the form shown in Figure 3 and is invariant under the one-dimensional subspace generated by (0, 1, 1, 1, 1). Hence each of the six vertices represents a two-dimensional cone and each of the five edges represents a three-dimensional cone.
Intersected with the affine hyperplane {−1} × R 4 , we obtain a polyhedral complex as shown in Figure 4, any vertex of Figure 3 in {0}×R 4 becoming a point at infinity.

Tracing Gröbner complexes to a trivial valuation
In this section, we show how the Gröbner complexes of ideals in K[x] can be traced back to the Gröbner fans of ideals in R t [x]. We will show how the Gröbner fan induces a refinement of the Gröbner complex and how to determine whether two integral Gröbner cones map to the same valued Gröbner polytope. For the latter, we will need to delve into some basics in Gröbner bases. We close this section with a remark on p-adic Gröbner bases as introduced by Chan and Maclagan [ChM13].
Definition 3.1 (Gröbner polyhedra, Gröbner complexes over valued fields) For a homogeneous ideal I K[x] and a weight vector w ∈ R n we define its Gröbner polytope to be where (·) denotes the closure in the euclidean topology. We will refer to the collection Σ ν (I) := {C ν,w (I) | w ∈ R n } as the Gröbner complex of I.
be a homogeneous ideal. Then all C ν,w (I) are convex polytopes and Σ ν (I) is a finite polyhedral complex.

Definition 3.3
For an x-homogenous ideal I R t [x], i.e. an ideal generated by elements which are homogeneous if considered as polynomials in x with coefficients in R t , and a weight vector w ∈ R <0 × R n we define its Gröbner cone to be where (·) denotes the closure in the euclidean topology. We will refer to the collection Σ(I) : be an x-homogeneous ideal. Then all C w (I) are polyhedral cones and Σ(I) is a finite polyehdral fan.
Proof. Follows directly from Proposition 2.8.
Note that it may very well happen that several cones are mapped into the same Gröbner polytope, i.e. that the image of the restricted Gröbner fan is a refinement of the Gröbner complex (see Example 3.10). We will now recall the notion of initially reduced standard bases of ideals in R t [x] from [MRW17] and how they determine the inequalities and equations of Gröbner cones as shown in [MaR17]. We will then use them to decide whether two Gröbner cones are mapped to the same Gröbner polytope and, by doing so, show that no separate standard basis computation is required for it.
Definition 3.6 (initially reduced standard bases) Fix the t-local lexicographical ordering > such that x 1 > . . . > x n > 1 > t. Given a weight vector w ∈ R <0 × R n we define the weighted ordering > w to be For g ∈ R t [x], the leading term LT >w (g) is the unique term of g with maximal monomial under > w and for I R t [x], the leading ideal LT >w (I) is the ideal generated by the leading terms of all its elements. A finite subset G ⊆ I is called a standard basis of I with respect to > w , if the leading terms of its elements generate LT >w (I).
is reduced in the classical sense. be an x-homogeneous ideal, let w ∈ R <0 × R n be a weight vector and let G an initially reduced standard basis of I with respect to > w . Then the set of its initial forms {in w (g) | g ∈ G} is an initially reduced standard basis of in w (I) with respect to > w , and the Gröbner cone of I around w is given by We now show that our standard bases of π −1 I R t [x] yield Gröbner bases of initial ideals of I K[x], allowing us to immediately decide whether two Gröbner cones of the former are mapped to the same Gröbner polytope of the latter.
Corollary 3.9 Let I K[x] be a homogeneous ideal, let w ∈ R n be a weight vector and let G be an initially reduced standard basis of π −1 I with respect to the weighted ordering > (−1,w) . Then is a standard basis of in ν,w (I) with respect to the fixed lexicographical ordering > restricted to monomials in x.
Remark 3.11 (homogenization and dehomogenization) A lot of effort has been put into developing algorithms for computing Gröbner cones C w (I) for x-homogeneous ideals I R t [x] and weight vectors w ∈ R <0 × R n in [MaR17] which terminate in finite time in case I can be generated by elements in and hence is restricted to the positive orthant R 4 ≥0 . However, once homogenized it yields a regular Gröbner fan Σ(I h ) living in R 5 , whose restriction to {0} × R 4 ≥0 refines Σ(I).
Remark 3.12 (p-adic Gröbner bases) A Gröbner basis of an ideal I K[x] over valued fields with respect to a weight vector w ∈ R n is by [MaS15, Section 2.4] a finite generating set whose initial forms generate the initial ideal in ν,w (I). Observe that Corollary 3.9 implies that such a Gröbner basis can be computed by projecting an initially reduced standard basis of π −1 I R t [x] under the monomial ordering > w via π to K[x]. Λ := (β, α) ∈ N × N n α ∈ N n with c α,β ′ = 0 for some β ′ ∈ N β = min{β ′ ∈ N | c α,β ′ = 0} .
The computation of general tropical varieties on the other hand works in three steps: (1) Finding a first maximal Gröbner cone C w (I) ⊆ T (I), Alg. 4.7.
of which (3) is a generalisation of the well-known flip of Gröbner bases, which we will simply cite from [MaR17] without going into any algorithmic details: • > w a weighted monomial ordering with weight vector w ∈ R <0 × R n , • v an outer normal vector of C w (I), • G = {g 1 , . . . , g k } ⊆ I an initially reduced standard basis of an x-homogeneous ideal I w.r.t. > w , To show how to find a first maximal dimensional Gröbner cone on T (I), we need to introduce the homogeneity space, since the starting cone algorithm works inductively over the codimension of it, and we have to recall the lift of standard bases, which we will again cite from [MaR17] without going into any algorithmic details. The latter allows us to lift a standard basis of an initial ideal into a standard basis of the original ideal, useful for avoiding unnecessary standard basis computations.

Definition 4.3 (homogeneity space)
Given an x-homogeneous ideal I R t [x], we define the homogeneity space of I (or of T (I)) to be the intersection of all its lower Gröbner cones, i.e. Gröbner cones of the form C w (I) for some w ∈ R <0 × R n , C 0 (I) := w∈R <0 ×R n C w (I).

Example 4.4
Note that our definition of homogeneity space C 0 (I) differs from the natural lineality space C 0 (I) of tropical varieties over fields with trivial valuation. In general, our C 0 (I) is neither a linear subspace nor is it the set of all vectors with respect to whom the ideal is weighted homogeneous. Consider the principal ideal whose Gröbner Fan splits the weight space R ≤0 × R 2 into two maximal cones, see Figure 6, and whose homogeneity space is given by Clearly, C 0 (I) is no subspace and we have (−1, 0, 0) ∈ C 0 (I) despite the ideal not being weighted homogeneous with respect to it. This effect is caused by the terms tx and ty in the generator, which do not appear in any initial form and hence have no influence on C 0 (I), yet still exist and thus prevent I from being weighted homogeneous with respect to any weight vector in the interior of C 0 (I).
We follow up our observation in Example 4.4 with the following Lemma, which shows that the homogeneity space behaves properly in the case which is of interest to us: be an x-homogeneous ideal and w ∈ R <0 × R n a weight vector. Then Proof. The second equality follows directly from the perturbation of initial ideals, i.e. it follows from the fact that for any v ∈ R <0 ×R n we have in v in w (I) = in w+ε·v (I) for ε > 0 sufficiently small [MaR17, Proposition 5.4]. It remains to show the first equality. The ⊇ inclusion can be shown in a similar fashion: Suppose v ∈ R <0 × R n such that in v (in w (I)) = in w (I). Then for any u ∈ R <0 × R n we have in v+ε·u (in w (I)) = in u (in v (in w (I))) = in u (in w (I)), showing that v + ε · u ∈ C u (in w (I)) for any ε > 0 sufficiently small. As C u (in w (I)) is closed by definition, this implies v ∈ C u (in w (I)). This shows that v is contained in every lower Gröbner cone of in w (I), and hence also in their intersection C 0 (in w (I)). For the ⊆ inclusion, consider v ∈ C 0 (in w (I)) ∩ (R <0 × R n ), so that v ∈ C u (in w (I)) for all u ∈ R <0 × R n . In particular, v ∈ C w (in w (I)) which is the middle set by definition.  ideal I with respect a weighted ordering > w , w ∈ R <0 × R n . Output: (C w ′ (I), G ′ , > w ′ ), where C w ′ (I) ⊆ T (I) maximal dimensional and G ′ an initially reduced standard basis of I with respect to the weighted ordering > w ′ . 1: if dim(I) = dim(C 0 (I)) then return (C 0 (I), G, >) 2: Find a weight vector w ∈ (T (I) \ C 0 (I)) ∩ (R <0 × R n ). 3: Compute an initially reduced standard basis G ′′ of I with respect to > w . 4: Set H ′′ := {in w (g) | g ∈ G ′′ }. 5: Rerun (C w ′ 0 (I), G ′ 0 , > ′ 0 ) = TropStartingCone(H ′′ , > w ). 6: Let > ′ be the weighted ordering with weight vector w and tiebreaker > ′ 0 . 7: Lift G ′ 0 to an initially reduced standard basis G ′ of I: A short calculation reveals that dim(T (I)) = dim(I) = 3 > 1 = dim(C 0 (I)) with C 0 (I) = R · (0, 1, 1, 1, 1).
Two centrals tools necessary to describe the tropical variety around one of its codimension one cells are generic weight vectors and tropical witnesses.
Definition 4.9 (multiweights and generic weights) Given weight vectors w ∈ R <0 × R n and v 1 , . . . , v d ∈ R × R n , we define the initial form of an element g ∈ R t [x] with respect to the multidegree (w, v 1 , . . . , v d ) to be and we define the initial ideal of I R t [x] with respect to (w, v 1 , . . . , v d ) to be Also, still fixing the lexicographical ordering > with x 1 > . . . > x n > 1 > t from Definition 3.6, we define the multiweighted ordering > (w,v 1 ,...,v k ) to be Moreover, given a polyhedral cone σ ⊆ R ≤0 × R n of dimension d with σ {0} × R n and a point w ∈ relint(σ) (note that w ∈ R <0 × R n necessarily), we call a weight vector u ∈ σ generic around w, if for all open neighbourhoods U around w there exists a weight vector u ′ ∈ U ∩ σ not lying on any Gröbner cone of dimension lower than d such that in u ′ (π −1 I) = in u (π −1 I).
Algorithm 4.10 (in σ,w (G), generic initial ideal around a weight)

Definition 4.11
Let I R t [x] and let u ∈ R <0 × R n be such that C u (I) T (I). We call an element f ∈ I a tropical witness of C u (I) if in v (f ) is a monomial for all v ∈ Relint(C u (I)). (1) > w a weighted monomial ordering for some w ∈ R <0 × R n , (2) G = {g 1 , . . . , g k } an initially reduced standard basis of an x-homogeneous ideal I R t [x] with respect to > w , (4) m ∈ in w (I) a monomial.
Output: f ∈ I, a tropical witness of C w (I).

15:
Construct the adjacent Gröbner cone As in w (2 − t) = 2 for all w ∈ R <0 × R 3 , it suffices to solely focus on g. For the starting cone, we begin with weight vector w = (−3, −10, 1, 0) ∈ R <0 × R 3 , since in w (g) = xy + t 3 z 2 is no monomial. In fact, its initial form is binomial, hence the only weight vectors v such that in w+εv (g) is no monomial are the v such that in w+εv (g) = in w (g), or in other words v ∈ C w (I). This shows that C w (I) is a maximal cone in the tropical variety. Note that all Gröbner cones are invariant under translation by (0, 1, 1, 1). Hence the 3-dimensional Gröbner cone C w (I) is spanned by two rays, which are generated by v 1 = (−2, −7, 1, 0) and v 2 = (−1, −3, 0, 0) respectively. This can be seen from their respective initial forms, which gain one additional term compared to in w (g), in v 1 (g) = xy + t 4 y 2 + t 3 z 2 and in v 2 (g) = xy + xz + t 3 z 2 . We have thus finished computing a starting cone and identified its two facets, which we need to traverse.
Continuing with direction v 1,2 = (0, 0, 1, 1), to whose side lies the closure of equivalence class such that in w ′ (g) = t 4 y 2 + t 3 z 2 , we see that the other ray of the maximal Gröbner cone is generated by v 3 = (0, 0, 1, 1) with in v 3 (g) = t 4 y 2 + t 3 z 2 . The ray lies on the boundary of the maximal Gröbner cone because it lies on the boundary of the lower halfspace. Continuing with the direction v 1,1 = (0, 0, 0, −1), which is the closure of the equivalence class such that in w ′ (g) = xy + t 4 y 2 , we get that the other ray of the maximal Gröbner cone is v 4 = (0, 0, 0, −1) with in v 4 (g) = t 2 x 2 + xy + t 4 y 2 . Because both v 3 and v 4 lie on the boundary of the lower halfspace, the only facet left to traverse is the one generated by v 2 . The tropical star around v 2 consists of three rays. One ray points in the direction of v 2,1 = (0, 1, 0, 0) so that in v 2 +ε·v 2,1 (g) = xy + xz. Another ray points in the direction of v 2,2 = (0, 0, −1, 0) so that in v 2 +ε·v 2,2 (g) = xz + t 3 z 2 . The final ray points in the direction of v 2,3 = (0, 0, 2, 1) so that in v 2 +ε·v 2,3 (g) = xy + t 3 z 2 = in w (g), this is the vector pointing into our starting cone.
Because v 6 lies on the boundary of the lower halfspace, v 5 generates the only facet left to traverse. A quick glance at the initial forms imply that it is connected to the facets generated by v 4 and v 6 , as it has two terms in common with each of them.
We obtain that T (I) is covered by a polyhedral fans which, modulo the homogeneity space R · (0, 1, 1, 1), has 6 rays, of which the ones generated by v 1 , v 2 , v 5 lie in the interior of the lower halfspace R ≤0 × R n , while the ones generated by v 3 , v 4 , v 6 lie on its boundary. The 6 rays are pairwise connected via 7 edges. The edges connecting (v 1 , v 3 ), (v 1 , v 4 ), (v 2 , v 6 ) and (v 4 , v 5 ) intersect the boundary in codimension one, while the cones connecting (v 1 , v 2 ) and (v 2 , v 5 ) intersect the boundary in codimension 2, which has to be the homogeneity space.
. Figure 12 shows its tropical varieties for all possible valuations on Q. Regardless of the valuation, all tropical varieties share the same recession fan, as was proven by Gubler [Gub13]. The latter is also necessarily the tropical variety under the trivial valuation. Note that for p sufficiently large, the tropical varieties under ν p coincides with the tropical variety under the trivial valuation. This is because p is simply too large for p − t to matter in any of our standard basis calculations. These p are referred to as good primes while other p are referred to as bad primes in the theory of modular techniques [BDF + 16].
Unlike Example 4.18, its tropical variety does not seem to dependent on the choice of valuation, which is not surprising as Speyer and Sturmfels showed that it is characteristic-free [SpS04, Theorem 7.1]. In this case, the computations under the p-adic valuation are mathematically equivalent to the computations under the trivial valuation, though the practical timings under the p-adic valuation are slightly slower due to a constant overhead of a more general framework. Figure 13 shows a shortened output of Singular when computing its tropical variety with respect to the 2-adic valuation. It describes a polyhedral fan whose intersection with the affine hyperplane {−1} × R 10 yields again a polyhedral fan: The ray #0 represents the 5-dimensional lineality space of T ν 2 (I), while the maximal cones {0 i j} represent polyhedral cones in T ν 2 (I) spanned by the lineality space and rays #i, #j. Note that, from a perspective of R n = {−1} × R n , all data is given in homogenized coordinates, which is why the f-Vector shown is slightly distorted by lower-dimensional cones at infinity. Figure 14 illustrates the combinatorial structure of ∆. Each vertex represents a ray of ∆, while each edge represents a maximal cone of ∆. The graph shown should be thought of as lying on a sphere S 2 , on which the colored edges connect with their counterpart on the other side.

Optimizations for non-trivial valuations
Up till now, all algorithms for computing T ν (I) via T (π −1 I) appear to be strictly worse than computing T (I), as we are working with an inhomogeneous ideal π −1 I over a coefficient ring R instead of a homogeneous ideal I over a coefficient field K.
Convention 5.1 Let I K[x] be a homogeneous ideal and fix an initial ideal J := in (−1,w) (π −1 I) R[t, x] of its preimage as well as the corresponding monomial ordering > (−1,w) . Note that necessarily p ∈ J.
Lemma 5.2 (quasi-homogeneity of J) There exists a positive weight vector u ∈ (R >0 ) n+1 such that J is weighted homogeneous with respect to it.