Computing Tropical Varieties Over Fields with Valuation
 100 Downloads
Abstract
We show how the tropical variety of an ideal \(I\unlhd K[x_1,\ldots ,x_n]\) over a field K with nontrivial discrete valuation can always be traced back to the tropical variety of an ideal \(\pi ^{1}I\unlhd R\llbracket t\rrbracket [x_1,\ldots ,x_n]\) over some dense subring R in its ring of integers. We show that this connection is compatible with the Gröbner polyhedra covering them. Combined with previous works, we thus obtain a framework for computing tropical varieties over general fields with valuations, which relies on the existing theory of standard bases if \(\pi ^{1}I\) is generated by elements in \(R[t,x_1,\ldots ,x_n]\).
Keywords
Tropical geometry Tropical varieties Gröbner fans Standard bases Valued fieldsMathematics Subject Classification
14T05 13P10 13F25 12J251 Introduction
Given a polynomial ideal I over a field K with a nontrivial valuation \(\nu :K \rightarrow {\mathbb {R}}\cup \{\infty \}\), its tropical variety \({{\,\mathrm{{\mathcal {T}}}\,}}(I)\) is commonly described as the combinatorial shadow of its vanishing set over the algebraic closure of K. Tropical varieties arise naturally in many contexts in mathematics [1, 22] and beyond, such as phylogenetic trees in biology [24, §4], productmix auctions in economics [2, 32] or finiteness of central configurations in the 5body problem in physics [12].
However, 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, Sturmfels and Thomas [5] for the rational function field over the complex numbers \({\mathbb {C}}(t)\) using classical Gröbner basis methods. More recently, Chan and Maclagan [7] generalised the notion of Gröbner bases to general fields with valuation in order to compute tropical varieties thereover. The linchpin of both works is the ability to compute initial ideals. Moreover, significant advances have been made in specific parts of the computations: Chan [6], Hofmann and Ren [13], Sommars and Verschelde [29] all worked on improving the main bottleneck that is the computation of tropical links. The first two works developed new algorithms based on projections and intersections, respectively, whereas the latter improved the computation of socalled tropical prevarieties which was essential for the original algorithm. At the same time, Vaccon [34] showed that MatrixF5 ideas can be applied to improve the performance of the generalised Gröbner bases computation.
In contrast, this article revisits the problem on a more fundamental level. As in [7], the overall goal is to develop a framework for general fields with valuation in which the original algorithms in [5] work almost ad verbum. However, instead of introducing a new notion of Gröbner bases, we aim to base it on the existing theory of standard bases [25]. The key idea is to use Cohen’s Structure Theorem and replace the valued field K with a power series ring \(R\llbracket t\rrbracket \) with its natural valuation. This replaces the original ideal \(I\unlhd K[x]\) with an ideal in \({R\llbracket t\rrbracket [x]}\), which is generated by polynomials in both t and x under mild assumptions on I. Our approach is to a certain extent equivalent to that of Chan and Maclagan, which can be seen from the fact that we naturally obtain an algorithm for computing their Gröbner bases. However, we can leverage existing implementations, such as in the computer algebra system Singular [8], for a better performance (see Timings 1).
Our framework relies heavily on two previous works: In [19], we introduced standard bases for ideals in \({R\llbracket t\rrbracket [x]}\), whose elements are multivariate polynomials in x and univariate power series in t over a coefficient ring R. In [20], we introduced Gröbner fans for ideals in \({R\llbracket t\rrbracket [x]}\), which are a natural amalgamation of the existing notions of Gröbner fans for power series rings [3, 26, 33] and Gröbner fans for polynomial rings over coefficient rings [23]. In both works, special emphasis is put on ideals in \({R\llbracket t\rrbracket [x]}\) generated by polynomials in t and x. For those, our standard bases coincide with the existing notion of standard bases for polynomials over coefficient rings and our algorithms consist of a finite sequence of basic polynomial arithmetic.
This article is organised as follows: First, in Sect. 2, we recall Cohen’s Structure Theorem and use it to establish a bijection between the tropical variety of an ideal in K[x] and the tropical variety of a corresponding ideal in \({R\llbracket t\rrbracket [x]}\) (see Theorem 4). Next, in Sect. 3, we show that this bijection is compatible with the polyhedral structures covering the respective tropical varieties (see Corollary 3). Finally, in Sect. 4, we explain how the corresponding ideal in \({R\llbracket t\rrbracket [x]}\) can be computed from the original ideal in K[x].
Furthermore, modified versions of the algorithms in [5] in our framework have been implemented in the Singular library tropical.lib [15], relying on gfanlib [14, 16] for computations in convex geometry (see Example 5). They are publicly available as part of the official Singular distribution, and a detailed account on the modified algorithms can be found in [27].
2 Tracing Tropical Varieties to a Trivial Valuation
The aim of this section is to show how tropical varieties over a valued field K can be traced back to tropical varieties over a power series ring \(R\llbracket t\rrbracket \) as in Convention 2. The linchpin is to show how initial ideals over one can be described through initial ideals over the other, and the remaining results then follow naturally from this. Let us start by recalling Cohen’s Structure Theorem.stop
Theorem 1
Convention 2
Example 1
Example 2
 1.
\(K=k((t))\) the field of Laurent series over a field k with \({\mathcal {O}}_K=k\llbracket t \rrbracket \) the ring of power series over k, \(R=k[t]\) and \(p=t\); e.g. \(k={\mathbb {F}}_q\) with q a prime power, as used in [30, Sect. 7] or [17], or \(k={\mathbb {Q}}\) as considered in [5] (see Example 4).
 2.
Finite extensions K of \({\mathbb {Q}}_p\) and \({\mathbb {F}}_q((t))\), i.e. all local fields with nontrivial valuation, and also all higherdimensional local fields.
 3.
\({\mathcal {O}}_K\) any completion of a localisation of a Dedekind domain R at a prime ideal \(P\unlhd R\), \(p\in P\) a suitable element. Note that p does not need to generate P and hence \({\mathcal {O}}_K\) need not be the completion with respect to the ideal generated by p, e.g. \(R={\mathbb {Z}}[\sqrt{5}]\), \(P=\langle 2,1+\sqrt{5}\rangle \) and \(p=2\).
 4.
For an odd choice of R, consider \(K:={\mathbb {Q}}(s)((t))\) so that \({\mathcal {O}}_K={\mathbb {Q}}(s)\llbracket t\rrbracket \). Set \(R:=S^{1}{\mathbb {Q}}[s,t]\), where \(S:={\mathbb {Q}}[s,t]\setminus (\langle t1,s\rangle \cup \langle t\rangle )\) is multiplicatively closed as the complement of two prime ideals. Then R is a noncatenarian, dense subring of \({\mathcal {O}}_K\).
To fix the notation, we briefly recall some basic notions in tropical geometry that are of immediate relevance to us. For an indepth introduction to tropical geometry, we refer to the reader to [18]. For a brief survey with a view towards algebraic geometry, we recommend [9].
Definition 1
Theorem 3
(Structure Theorem for Tropical Varieties, [18, Theorem 3.3.5]) Let \(I\unlhd K[x]\) define an irreducible subvariety in \((K^*)^n\) of dimension d. Then \({{\,\mathrm{{\mathcal {T}}}\,}}_{\nu }(I)\) is the support of a pure polyhedral complex of the same dimension that is connected through codimension 1.
Next, we introduce initial forms and initial ideals in for elements and ideals in \({R\llbracket t\rrbracket [x]}\) and show how initial ideals of ideals in \({R\llbracket t\rrbracket [x]}\) can be used to compute the initial ideals of ideals in K[x].
Definition 2
Example 3
Proposition 1
Proof
 \(\supseteq \)
 Any term \(s\in {\mathcal {O}}_K[x]\) is of the form \(s=(\sum _\beta c_\beta p^\beta )\cdot x^\alpha \) with \(p\not \mid c_\beta \) for all \(\beta \in {\mathbb {N}}\). Then the element \(s':=(\sum _\beta c_\beta t^\beta )\cdot x^\alpha \in {R\llbracket t\rrbracket [x]}\) is a natural preimage of s under \(\pi \) for which we haveAnd because the valued weighted degree of s in \({\mathcal {O}}_K [x]\), i.e. the lefthand side in the following equation, and the weighted degree of \(s'\) in \({R\llbracket t\rrbracket [x]}\), i.e. the righthand side of the following equation, coincide,$$\begin{aligned} {{\,\mathrm{in}\,}}_{\nu ,w}(s)=\overline{c}_{\beta _0} \cdot x^\alpha =\overline{{{\,\mathrm{in}\,}}_{(1,w)}({s'})}_{t=1}, \text { where } \beta _0=\min \{\beta \in {\mathbb {N}}\mid c_\beta \ne 0\}. \end{aligned}$$this implies that any \(f\in {\mathcal {O}}_K[ x]\) has a preimage \(f'\in {R\llbracket t\rrbracket [x]}\) under \(\pi \) such that$$\begin{aligned} \deg _w(x^\alpha )\nu (\textstyle \sum _\beta c_\beta p^\beta ) = \max \{w\cdot \alpha  \beta \;\;c_\beta \not =0\} = \deg _{(1,w)}(\textstyle \sum _\beta c_\beta \cdot t^\beta x^\alpha ), \end{aligned}$$simply by applying the above argument to each of its terms.$$\begin{aligned} {{\,\mathrm{in}\,}}_{\nu ,w}(f)=\overline{{{\,\mathrm{in}\,}}_{(1,w)}({f'})}_{t=1}, \end{aligned}$$
 \(\subseteq \)

Once again consider a term \(s=\sum _\beta c_\beta p^\beta \cdot x^\alpha \in {\mathcal {O}}_K[x]\) with \(p\not \mid c_\beta \) for all \(\beta \in {\mathbb {N}}\). Then any preimage of it under \(\pi \) is of the form \(s'=\sum _\beta c_\beta t^\beta x^\alpha + r\) for some \(r\in \langle pt\rangle \).
If \(\deg _{(1,w)}(r) > \deg _{(1,w)}(\sum _\beta c_\beta t^\beta x^\alpha )\), we havesince \({{\,\mathrm{in}\,}}_{(1,w)}(r)\in {{\,\mathrm{in}\,}}_{(1,w)}\langle pt\rangle = \langle p\rangle \). And if \(\deg _{(1,w)}(r) < \deg _{(1,w)}(\sum _\beta c_\beta t^\beta x^\alpha )\), we have$$\begin{aligned} \overline{{{\,\mathrm{in}\,}}_{(1,w)}({s'})}_{t=1}=\overline{{{\,\mathrm{in}\,}}_{(1,w)}({r})}_{t=1}=0, \end{aligned}$$where \(\beta _0:=\min \{\beta \in {\mathbb {N}}\mid c_\beta \ne 0\}\). Now suppose \(\deg _{(1,w)}(r) = \deg _{(1,w)}(\sum _\beta c_\beta t^\beta x^\alpha )\). First observe that because t is weighted negatively, there can be no cancellation amongst the highest weighted terms of r and the terms of \(\sum _\beta c_\beta t^\beta x^\alpha \), as the terms of \(\sum _\beta c_\beta t^\beta x^\alpha \) are not divisible by p, unlike the terms of the highest weighted terms of r. Therefore, we have$$\begin{aligned} \overline{{{\,\mathrm{in}\,}}_{(1,w)}({s'})}_{t=1}&=\overline{{{\,\mathrm{in}\,}}_{(1,w)}(\textstyle \sum _\beta c_\beta t^\beta x^\alpha )}_{t=1} =\overline{c}_{\beta _0} \cdot x^\alpha \\&={{\,\mathrm{in}\,}}_{\nu ,w}(\textstyle \sum _\beta c_\beta p^\beta \cdot x^\alpha )= {{\,\mathrm{in}\,}}_{\nu ,w}(s), \end{aligned}$$Either way, we always have \(\overline{{{\,\mathrm{in}\,}}_{(1,w)}({s'})}_{t=1} \in \langle {{\,\mathrm{in}\,}}_{\nu ,w}(s)\rangle \) for any arbitrary preimage \(s'\in \pi ^{1}(s)\), and, as before, the same hence holds true for any arbitrary element \(f\in {\mathcal {O}}_K[ x]\). \(\square \)$$\begin{aligned} \overline{{{\,\mathrm{in}\,}}_{(1,w)}({s'})}_{t=1}&=\underbrace{\overline{{{\,\mathrm{in}\,}}_{(1,w)}(\textstyle \sum _\beta c_\beta t^\beta x^\alpha )}_{t=1}}_{={{\,\mathrm{in}\,}}_{\nu ,w}\textstyle (\sum _\beta c_\beta p^\beta \cdot x^\alpha )} + \underbrace{\overline{{{\,\mathrm{in}\,}}_{(1,w)}({r})}_{t=1}}_{=\overline{0}} ={{\,\mathrm{in}\,}}_{\nu ,w}(s). \end{aligned}$$
Corollary 1
Proof
The statement follows from \({{\,\mathrm{in}\,}}_{\nu ,w}(I)={{\,\mathrm{in}\,}}_{\nu ,w}(I\cap \mathcal O_K[ x])\) and Proposition 1. \(\square \)
Finally, we can introduce tropical varieties in \({R\llbracket t\rrbracket [x]}\) and show how they relate to tropical varieties in K[x]. In particular, we note how the tropical varieties in \({R\llbracket t\rrbracket [x]}\) that are of interest to us are pure and connected through codimension 1. This is not a given for tropical varieties over coefficient rings [18, §1.6] and very important algorithmically, as it allows us to run over it via a fantraversal through the facets of the maximal cones.
Definition 3
Theorem 4
Proof
 \(\Longrightarrow \)

Suppose \({{\,\mathrm{in}\,}}_{(1,w)}(\pi ^{1} I)\unlhd {R\llbracket t\rrbracket [x]}\) contains a monomial \(t^\beta x^\alpha \). By Corollary 1, we have \({{\,\mathrm{in}\,}}_{\nu ,w}(I) = \overline{{{\,\mathrm{in}\,}}_{(1,w)}({\pi ^{1} I})}_{t=1}\), which means \({{\,\mathrm{in}\,}}_{\nu ,w}(I)\) must contain the monomial \(x^\alpha \in {\mathfrak {K}}[ x]\).
 \(\Longleftarrow \)
 Suppose \({{\,\mathrm{in}\,}}_{\nu ,w}(I)\unlhd {\mathfrak {K}}[x]\) contains a monomial \(x^\alpha \). For the remainder of the proof, we abbreviate \((1,w)\)weighted degree and \((1,w)\)weighted homogeneous with weighted degree and weighted homogeneous, respectively. Consider all \(r\in R[t,x]\) such that$$\begin{aligned} f:=t^\beta \cdot \big (x^\alpha +(t1)\cdot r\big )\in {{\,\mathrm{in}\,}}_{(1,w)}(\pi ^{1} I), \text { for some } \beta \in {\mathbb {N}}. \end{aligned}$$
Hence, \(q_l\le d< q_1\), which, however, contradicts \(q_i<q_{i+1}\). \(\square \)
Corollary 2
If \(I\unlhd K[x]\) defines an irreducible subvariety of \((K^*)^n\) of dimension d, then \({{\,\mathrm{{\mathcal {T}}}\,}}(\pi ^{1} I)\) is the support of a pure polyhedral fan of dimension \(d+1\) connected through codimension one.
Proof
Follows immediately from Definition 3 and Theorem 4, which imply that \({{\,\mathrm{{\mathcal {T}}}\,}}(\pi ^{1} I)\) is the polyhedral fan over \({{\,\mathrm{{\mathcal {T}}}\,}}_\nu (I)\). And by Theorem 3, the latter is pure of dimension d and connected through codimension one. \(\square \)
We close the section with a couple of examples of tropical varieties over K[x], their counterparts in \({R\llbracket t\rrbracket [x]}\) and how they can be computed in Singular.
Example 4
Example 5
Intersecting with the affine hyperplane \(\{1\}\times {\mathbb {R}}^4\), we obtain a polyhedral complex as shown in the top left of Fig. 3, the vertices of Fig. 2 in \(\{0\}\times {\mathbb {R}}^4\) becoming points at infinity.
3 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 \({R\llbracket t\rrbracket [x]}\). We show how the latter induces a refinement of the former and how to determine whether two Gröbner cones map to the same Gröbner polyhedron. We close this section with a remark on padic Gröbner bases as introduced by Chan and Maclagan [7].
Definition 4
Theorem 5
([18, Theorem 2.5.3]) Let \(I\unlhd K[x]\) be a homogeneous ideal. Then all \(C_{\nu ,w}(I)\) are convex polytopes and \(\Sigma _\nu (I)\) is a finite polyhedral complex.
Definition 5
Proposition 2
([20, Theorem 3.19]) Let \(I\unlhd {R\llbracket t\rrbracket [x]}\) be an xhomogeneous ideal. Then all \(C_w(I)\) are polyhedral cones and \(\Sigma (I)\) is a finite polyhedral fan.
Corollary 3
The map \(\{1\}\times {\mathbb {R}}^n{\mathop {\longrightarrow }\limits ^{\sim }}{\mathbb {R}}^n, (1,w)\longmapsto w\) is compatible with the Gröbner fan \(\Sigma (\pi ^{1}I)\) and the Gröbner complex \(\Sigma _\nu (I)\), i.e. it maps the restriction of a Gröbner cone \(C_{(1,w)}(\pi ^{1}I)\cap \big (\{1\}\times {\mathbb {R}}^n\big )\) into the Gröbner polytope \(C_{\nu ,w}(I)\).
Proof
Follows directly from Proposition 1, as two weight vectors with the same initial ideal of \(\pi ^{1}I\unlhd {R\llbracket t\rrbracket [x]}\) yield the same initial ideal of \(I\unlhd K[x]\). \(\square \)
Note that it may 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 6).
We now recall the notion of initially reduced standard bases of ideals in \({R\llbracket t\rrbracket [x]}\) from [19] and how they determine the inequalities and equations of Gröbner cones as shown in [20]. We then use these 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 this.
Definition 6
Proposition 3
([20, Algorithm 4.6]) Let \(I\unlhd {R\llbracket t\rrbracket [x]}\) be an xhomogeneous ideal and \(w\in {\mathbb {R}}_{<0}\times {\mathbb {R}}^n\) a weight vector. Then an initially reduced standard basis G of I with respect to \(>_w\) can be computed using a finite sequence of arithmetic operations in \({R\llbracket t\rrbracket [x]}\). Moreover, if I is generated by elements in R[t, x], then it can be computed using a finite sequence of arithmetic operations in R[t, x].
Proposition 4
We now show that our standard bases of \(\pi ^{1}I\unlhd {R\llbracket t\rrbracket [x]}\) yield Gröbner bases of initial ideals of \(I\unlhd K[x]\), allowing us to immediately decide whether two Gröbner cones of the former are mapped to the same Gröbner polytope of the latter.
Corollary 4
Proof
By Proposition 4, the set \({{\,\mathrm{in}\,}}_{(1,w)}(G):=\{{{\,\mathrm{in}\,}}_{(1,w)}(g)\mid g\in G\}\) is an initially reduced standard basis of \({{\,\mathrm{in}\,}}_{(1,w)}(\pi ^{1}I)\) with respect to \(>_{(1,w)}\). And because it is homogeneous with respect to the weight vector \((1,w)\), it is also an initially reduced standard basis with respect to >. By choice of >, the set \({{\,\mathrm{in}\,}}_{(1,w)}(G)_{t=1}\) remains a standard basis of \({{\,\mathrm{in}\,}}_{(1,w)}(\pi ^{1}I)_{t=1}\) with respect to the restriction of > to monomials in x. And since \(p\in {{\,\mathrm{in}\,}}_{(1,w)}(G)_{t=1}\), \(\overline{{{\,\mathrm{in}\,}}_{(1,w)}({G})}_{t=1}\) is a standard basis of \(\overline{{{\,\mathrm{in}\,}}_{(1,w)}({\pi ^{1}I})}_{t=1}\) with respect to the restriction of >. \(\square \)
Example 6
Consider the preimage \(\pi ^{1}I\unlhd {\mathbb {Z}}\llbracket t\rrbracket [x,y,z]\) of the ideal \(I=\langle 2y+x,z^2+y^2\rangle \unlhd {\mathbb {Q}}_2[x,y,z]\) and the two weight vectors \(w=(1,3,7), v=(1,10,5)\in {\mathbb {R}}^3\). Fix a lexicographical tiebreaker > with \(x>y>z>1>t\).
Remark 1
(padic Gröbner bases) By [18, Sect. 2.4], a Gröbner basis of an ideal \(I\unlhd K[x]\) over valued fields with respect to a weight vector \(w\in {\mathbb {R}}^n\) is a finite generating set whose initial forms generate the initial ideal \({{\,\mathrm{in}\,}}_{\nu ,w}(I)\). By Corollary 4, \(\pi (G)\) is such a Gröbner basis if \(G\subseteq \pi ^{1}I\unlhd {R\llbracket t\rrbracket [x]}\) is a standard basis under the monomial ordering \(>_w\).
Lines 1 to 6 in Fig. 5 illustrate the computation of a Gröbner bases over the 2adic numbers in Singular: Line 3 creates the Katsura(4) ideal in \(x_1,\ldots ,x_4\), Line 5 homogenises it on \(x_0\) and adds the generator \(2t\), and Line 6 computes its standard basis. Note that ds is a weighted ordering with weight vector \((1,\ldots ,1,1)\) which is equivalent to a weighted ordering with weight vector \((0,\ldots ,0,1)\) since the ideal is homogeneous in \(x_0,\ldots ,x_4\). Substituting t with 2 in stdI yields a Gröbner basis, however the monomials will be out of order since the ordering ignores the 2adic valuation.
Lines 7 to 12 in Fig. 5 construct the initial ideal: Line 7 forces Singular to do tail reductions even though this might cause infinite loops in nonglobal orderings. Line 8 reduces stdI with respect to \(2t\), so that the minimal degrees in t reflect the 2valuations. Line 9 computes the desired initial form, and Line 10 replaces all t with 1 so that Line 12 can safely pass to a polynomial ring without t over the residue field.
 Cyclic(n)

In \({\mathbb {Q}}_2[x_0,\ldots ,x_n]\), the cyclic ideal in the variables \(x_1,\ldots ,x_n\), homogenised using the variable \(x_0\), and weight vector \((0,\ldots ,0)\).
 Katsura(n)

In \({\mathbb {Q}}_2[x_0,\ldots ,x_n]\), the Katsura ideal in the variables \(x_1,\ldots ,x_n\), homogenised using the variable \(x_0\), and weight vector \((0,\ldots ,0)\).
 Chan

In \({\mathbb {Q}}_3[x_0,\ldots ,x_4]\), the ideal \(\langle 2x_1^2+3x_1x_2+24x_3x_4, 8x_1^3+x_2x_3x_4+18x_3^2x_4\rangle \) and weight vector \((1,11,3,19)\) taken from [6, §3.6].
Timings in seconds unless aborted after 1 CPU day
Examples  gfan  Macaulay2  Sage  Singular 

Cyclic(4)  –  1  10  1 
Cyclic(5)  –  –  –  1 
Cyclic(6)  –  –  –  2 
Katsura(3)  1  1  1  1 
Katsura(4)  –  –  10  1 
Katsura(5)  –  –  190  1 
Katsura(6)  –  –  2900  – 
Chan  1  1  4  – 
4 Computing the Preimage
This article was dedicated to show how \({{\,\mathrm{{\mathcal {T}}}\,}}_{\nu }(I)\) can be computed via \({{\,\mathrm{{\mathcal {T}}}\,}}(\pi ^{1}I)\), however until now we have not addressed how to determine the preimage \(\pi ^{1}I\) in the first place. We therefore end the article with two results: The first shows that \(\pi ^{1}I\) can be obtained by a saturation, and the second describes how to compute it.
Lemma 1
Proof
The \(\supseteq \) inclusion is obvious, as \(pt\) is mapped to 0 and p is invertible in K.
The next example shows that \(\langle f_1,\ldots ,f_k'\rangle \) in Lemma 1 is not necessarily saturated with respect to p, which is why Proposition 5 shows how to compute it.
Example 7
Consider \(I=\langle f_1,f_2 \rangle \unlhd {\mathbb {Q}}_2[x,y]\), where \(f_1=x^2+\frac{1}{2} y, f_2=y^2+\frac{1}{2}y\). Then \(g=x^2y^2\in I\cap {\mathbb {Z}}_2[x,y]\) and \(2g\in \langle f_1,f_2\rangle \unlhd {\mathbb {Z}}_2[x,y]\), but \(g\notin \langle f_1,f_2\rangle \unlhd {\mathbb {Z}}_2[x,y]\).
Lemma 2
Proof
Follows directly from Lemma 1, since \(pt\in \langle f_1',\ldots ,f_k' \rangle + \langle pt \rangle \) implies that its saturation with respect to p coincides with the saturation at t. \(\square \)
The following proposition shows how to compute the a standard basis of the preimage. It requires the notion of strong standard bases as in [25, Definition A.1.1.8]. The result and its proof is a straightforward generalisation of [31, Lemma 12.1].
Proposition 5
Let \(I \unlhd K[x]\) be an ideal, and let \(\{f_1,\ldots ,f_k\}\subseteq I\cap {\mathcal {O}}_K[x]\) be a generating set of I in the valuation ring. Since \(\pi :{R\llbracket t\rrbracket [x]}\rightarrow {\mathcal {O}}_K[x]\) is surjective, there are \(f_1',\ldots ,f_k'\in {R\llbracket t\rrbracket [x]}\) such that \(\pi (f_i')=f_i\in {\mathcal {O}}_K[x]\).
Proof
Example 8
Let I be the ideal Katsura(4) from Remark 1. Figure 6 shows the initially reduced standard basis of the computation in Fig. 5. By Proposition 5, dividing stdI[6] and stdI[7] by t yields a standard basis of the preimage. This shows that stdI does not generate the entire preimage in \({\mathbb {Z}}\llbracket t\rrbracket [x_1,\ldots ,x_4]\).
Notes
Acknowledgements
Open access funding provided by Max Planck Society.
References
 1.X. Allamigeon, P. Benchimol, S. Gaubert, and M. Joswig, Combinatorial simplex algorithms can solve mean payoff games, SIAM J. Optim. 24, no. 4, 2096–2117 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
 2.E. Baldwin and P. Klemperer, Understanding Preferences: ’Demand Types’, and The Existence of Equilibrium with Indivisibilities. Available at SSRN, https://doi.org/10.2139/ssrn.2643086 (2018)
 3.Rouchdi Bahloul and Nobuki Takayama, Local Gröbner fans, C. R. Math. Acad. Sci. Paris 344, no. 3, 147–152 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
 4.Janko Böhm, Wolfram Decker, Claus Fieker, Santiago Laplagne, and Gerhard Pfister, Bad primes in computational algebraic geometry, Mathematical Software – ICMS 2016, 93–101 (2016).MathSciNetzbMATHGoogle Scholar
 5.T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas, Computing tropical varieties, J. Symbolic Comput. 42, no. 12, 54–73 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
 6.Andrew Chan, Gröbner bases over fields with valuation and tropical curves by coordinate projections, Ph.D. thesis, University of Warwick (2013).Google Scholar
 7.Andrew J. Chan and Diane Maclagan, Gröbner bases over fields with valuations, Math. Comput. 88, no. 315, 467–483 (2019).CrossRefzbMATHGoogle Scholar
 8.Wolfram Decker, GertMartin Greuel, Gerhard Pfister, and Hans Schönemann, Singular 412 — A computer algebra system for polynomial computations, https://www.singular.unikl.de (2019).
 9.Andreas Gathmann, Tropical algebraic geometry, Jahresber. Dtsch. Math.Ver. 108, No. 1, 3–32 (2006).MathSciNetzbMATHGoogle Scholar
 10.Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, https://www.math.uiuc.edu/Macaulay2/ (2019).
 11.Walter Gubler, A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry, Contemp. Math. 589, 125–189 (2013).CrossRefzbMATHGoogle Scholar
 12.M. Hampton, and A. Jensen, Finiteness of spatial central configurations in the fivebody problem, Celestial Mech. Dynam. Astronom. 109, no. 4, 321–332 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
 13.Tommy Hofmann, Yue Ren, Computing tropical points and tropical links, Discrete Comput. Geom. 60, no. 3, 627–645 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
 14.Anders N. Jensen, Gfan 0.6.2, a software system for Gröbner fans and tropical varieties, http://home.math.au.dk/jensen/software/gfan/gfan.html (2017).
 15.Anders N. Jensen, Hannah Markwig, Thomas Markwig, and Yue Ren, tropical.lib. a Singular 412 library for computations in tropical goemetry (2019).Google Scholar
 16.Anders N. Jensen, Yue Ren, and Frank Seelisch, gfan.lib. A Singular 412 interface to gfanlib for basic computations in convex geometry (2019).Google Scholar
 17.Nikita Kalinin, Tropical approach to Nagata’s conjecture in positive characteristic, Discrete Comput. Geom. 58, no. 1, 158–179 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
 18.Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI (2015).CrossRefzbMATHGoogle Scholar
 19.Thomas Markwig, Yue Ren, and Oliver Wienand, Standard bases in mixed power series and polynomial rings over rings, J. Symbolic Comput. 79, 119–139 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
 20.Thomas Markwig and Yue Ren, Gröbner fans of xhomogeneous ideals in R \(\llbracket \) t \(\rrbracket \)[x], J. Symbolic Comput. 83, 315–341 (2017).Google Scholar
 21.Hideyuki Matsumura, Commutative Algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, (1980).Google Scholar
 22.G. Mikhalkin, Enumerative tropical algebraic geometry in \(\mathbb{R}^2\), J. Amer. Math. Soc. 18, no. 2, 313–377 (2005).Google Scholar
 23.T. Mora, and L. Robbiano, The Gröbner fan of an ideal, J. Symbolic Comput. 6, no. 23, 183–208 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
 24.L. Pachter, and B. Sturmfels, Algebraic statistics for computational biology, Cambridge Univ. Press, New York (2005).CrossRefzbMATHGoogle Scholar
 25.A. Popescu, Signature based standard bases over principal ideal rings, PhD Thesis, TU Kaiserslautern (2016).Google Scholar
 26.Patrick PopescuPampu and Dmitry Stepanov, Local tropicalization, Algebraic and combinatorial aspects of tropical geometry, Contemp. Math., vol. 589, 253–316 (2013).CrossRefzbMATHGoogle Scholar
 27.Y. Ren, Tropical geometry in Singular, Dissertation, Technische Universität Kaiserslautern, Germany (2015).Google Scholar
 28.Sage, Sagemath, the Sage Mathematics Software System (Version 8.7), https://www.sagemath.org (2019).
 29.Jeff Sommars and Jan Verschelde, Pruning algorithms for pretropisms of Newton polytopes, Computer Algebra in Scientific Computing – CASC 2016, 489–503 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
 30.David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4, no. 3, 389–411 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
 31.Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI (1996).zbMATHGoogle Scholar
 32.N. M. Tran, and J. Yu, ProductMix Auctions and Tropical Geometry, arXiv:1505.05737 (2015).
 33.Naoyuki Touda, Local tropical variety, arXiv:math/0511486 (2005).
 34.Tristan Vaccon, MatrixF5 algorithms over finiteprecision complete discrete valuation fields, J. Symbolic Comput. 80, Part 2, 329350 (2017).Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.