Computing Tropical Points and Tropical Links

We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical varieties, using intersections with affine hyperplanes to reduce the dimension to zero. We use the algorithms to show that the tropical Grassmannians G3,8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}_{3,8}$$\end{document} and G4,8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}_{4,8}$$\end{document} are not simplicial.


Introduction
Given an affine variety X over an algebraically closed field K with non-trivial valuation, its tropical variety Trop(X) is the euclidean closure of its image under component-wise valuation.Tropical varieties arise naturally in many applications in mathematics [ABGJ14;Mik05] and beyond, such as in the context of phylogenetic trees in biology [PS05, Section 4], product-mix auctions in economics [TY15] or finiteness of central configurations in the 5-body problem in physics [HJ11].
Nevertheless, computing tropical varieties is an algorithmically challenging task, requiring sophisticated techniques from computer algebra and convex geometry.The first algorithms were developed by Bogart, Jensen, Speyer, Sturmfels and Thomas [BJSST07] for the field of complex Puiseux series C{{t}}.More recently, Chan and Maclagan introduced a new notion of Gröbner bases for general fields with valuation in order to compute tropical varieties thereover [CM13].Concurrently, Chan developed a special algorithm for computing tropical curves [Cha13,Chapter 4].All these algorithms have been implemented in gfan [Jen17], which is the currently most widely used program for computing tropical varieties.In this article, we touch upon two problems that arise in the computation.
The first problem is to pinpoint a tropical starting point, a first point on the tropical variety from which all further computations start off.At present, the default is to traverse the Gröbner complex randomly while checking all vertices along the way for containment in the tropical variety.This is a rather inefficient approach however, as there can be significantly more Gröbner polyhedra outside the tropical variety than inside [BJSST07,Theorem 6.3].The second problem, which arises repeatedly, is to compute tropical links, tropical varieties of simpler combinatorial structure which describe the original tropical variety locally.Their special structure allows them to be computed via tropical prevarieties.While this has proven to be successful for a wide range of examples, experiments show that with increasing input size the tropical prevariety computations become intractable.
We present a simple yet novel approach for solving the aforementioned problems, based on the following bread-and-butter techniques in computer algebra and number theory: (1) intersection with random hyperplanes, (2) triangular decomposition of zero-dimensional polynomial ideals, (3) reading off valuations of roots from Newton polygons.
Moreover, the algorithm for tropical links also relies on a generalization of the Transverse Intersection Lemma [BJSST07, Lemma 3.2] to general fields with valuation, which follows from recent results by Osserman and Payne [OP13].
We use our algorithms to study some higher tropical Grassmannians G k,n .They were first studied by Speyer and Sturmfels [SS04], who showed that G 2,n for n ≥ 2 and G 3,6 are simplicial fans, the former using an intriguing connection to spaces of phylogenetic trees and the latter through explicit computation.Additionally, in their work on the parametrization and realizability of tropical planes [HJJS09], Hermannn, Jensen, Joswig and Sturmfels showed that G 3,7 is also a simplicial fan.We will complement these findings by showing that this does not hold for G 3,8 and G 4,8 .
All algorithms presented in this article have been implemented in the Singular library tropicalNewton.lib[DGPS16; HR16], and are publicly available as part of the official Singular distribution.For computations in convex geometry, it relies on an interface to gfanlib [Jen17; JRS16].

Convention 1.1
For the remainder of the article, let K be an algebraically closed field with nontrivial valuation ν : K → R ∪ {∞}, though we will mainly focus on its restriction ν : K * → R. We assume that 1 ∈ ν(K * ).As K is algebraically closed, there exists a homomorphism ψ : (ν(K * ), +) → (K * , •) with ν(ψ(w)) = w [MS15, Lemma 2.1.15].We will fix one such ψ and use p w to denote the element ψ(w) ∈ K * , or t w if K is the field of Puiseux series C{{t}}.Let K denote the residue field of K.
Furthermore, we fix a multivariate polynomial ring K[x] := K[x 1 , . . ., x n ].By abuse of notation, we will also use ν to refer to the component-wise valuation Acknowledgements.The authors would like to thank Michael Joswig and Benjamin Schröter for their feedback on a previous version of the article.

Computing zero-dimensional tropical varieties
In this section we present an algorithm, Algorithm 2.10, for computing zerodimensional tropical varieties using triangular decomposition and Newton polygon methods.For the sake of simplicity, we restrict ourselves to the task of computing a single point on the tropical variety, as the structure of the algorithm easily suggests how the entire tropical variety can be computed with proper bookkeeping.We conclude the section by showing that any generic triangular set admits what we call a tree of unique Newton polygons, which is the best case for our algorithm as it allows us to compute its tropical variety purely combinatorially, see Example 2.13.
we define the evaluation of its tropicalization at w to be and its initial form with respect to w to be For an ideal I K[x], we define its initial ideal with respect to w to be in The tropical variety of I is then given by For single polynomials f ∈ K[x] and finite subsets F ⊆ K[x], we abbreviate Trop(f ) := Trop( f ) and Trop(F ) := Trop( F ).The tropical variety is naturally covered by Gröbner polyhedra and hence the support of a subcomplex of the Gröbner complex [MS15, Theorem 3.3.2].Its dimension resp.lineality space is the dimension resp.lineality space of that subcomplex.
While the previous algorithms mainly work with the aforementioned definition of tropical varieties, the algorithms in this article focus on the following characterization: Theorem 2.2 ([MS15, Theorem 3.2.5])For any ideal I K[x] and its corresponding affine variety X = V (I) ⊆ K n we have where (•) denotes the closure in the euclidean topology.
We now describe how to exploit this geometric characterization algorithmically using triangular sets and Newton polygon methods.
Similarly, for a multivariate polynomial f = , and a weight w ∈ R k−1 , we define the expected Newton polygon of f at w to be We say f has a unique Newton polygon at w, if the initial form in w (f i ) is a monomial for all vertices (i, trop(f i )(w)) ∈ ∆ w (f ).Let Λ(f ) resp.Λ w (f ) denote the sets consisting of the negatives of the slopes of ∆(f ) resp.∆ w (f ).
The following two propositions justify the utility of Newton polygons and the term "unique Newton polygon".
Proposition 2.8 For a polynomial f ∈ K[x 1 , . . ., x k ] and a weight w ∈ R k−1 the following are equivalent: (1) f has a unique Newton polygon at w, (2) for all z ∈ K k−1 with ν(z) = w we have ∆(f (z, x k )) = ∆ w (f ).
Proof.Note that for any coefficient c ∈ K, any substitute z ∈ K k−1 with ν(z) = w and any exponent vector α ∈ N k−1 we have ν(c with equality guaranteed if in w (f i ) is a monomial, i.e. (1) implies (2).
Figure 1. the expected and possible Newton polygons of f .
Algorithm 2.10 (tropical point, zero-dimensional case only) if f i has a unique Newton polygon at (w 1 , . . ., w i−1 ) then

5:
else 6: Compute a root (z 1 , . . ., z i−1 ) ∈ V (f 1 , . . ., f i−1 ). 7: Proof.The termination of the algorithm is clear and the correctness follows directly from Propositions 2.7 and 2.8.While Algorithm 2.10 looks straightforward, performing Step 6 is a rather delicate task, which we will address in Examples 2.11 and 2.12.Example 2.13 shows how Algorithm 2.10 can be used to compute the entire tropical variety.
Example 2.11 (root approximation) Note that, in Step 6 of Algorithm 2.10, it always suffices to approximate the root with respect to the metric induced by the valuation.For instance, consider the triangular set (0, 0).However, f 3 does not have a unique Newton polygon at (0, 0) and ∆(f 3 (z 1 , z 2 , x 3 )) may vary depending on z 1 and z 2 .More precisely, we have Through Hensel Lifting we see that f 1 has a root z 1 ∈ Z 3 with z 1 ≡ 4 mod 3 2 Z 3 and f 2 (z 1 , x 2 ) has a root z 2 ∈ Z 3 with z 2 ≡ 1 mod 3 2 Z 3 .Since z 1 − z 2 = 0 and z 1 − z 2 ∈ 3Z 3 , we are in the second case and conclude that (0, 0, − 1 2 ) ∈ Trop(F ).
Example 2.12 (field extensions) While we began this article by fixing an algebraically closed field K, in practise we are always working over a finite extension of either the rationals Q, a finite field F q or function fields thereon.This can be problematic in conjunction with Step 6, as approximating roots might require further field extensions.By the recursive nature of the algorithm, we potentially end up with a tower of field extensions.For instance, consider the triangular set given by This triangular set will never encounter a unique Newton polygon in Step 3, and every root computation in Step 6 will require a new degree 2 extension, as This eventually leads to a degree 2 n extension of Q, which shows in the performance of our implementation of Algorithm 2.10 in tropicalNewton.lib:computing the tropicalization for n = 13 requires 8 seconds and it roughly doubles with each increment of n.See Timings 3.9 for a comparison with other algorithms.
Example 2.13 (computing entire tropical varieties) As mentioned in the beginning of the section, Algorithm 2.10 can be used to compute entire tropical varieties of zero-dimensional ideals.This is done by computing a triangular decomposition as in Proposition 2.4 and applying the algorithm to each triangular set, while exhausting all in Steps 4 and 6-7.For instance, consider the triangular set Then F admits several choices for slopes throughout the algorithm, and each choice in turn induces a new unique Newton polygon as illustrated in Figure 2. Keeping track of all of them, allows us to reconstruct its entire tropical variety: Figure 2. a tree of unique Newton polygons.
We conclude this section by showing that any generic triangular set resembles Example 2.13 in the sense that its tropical variety is determined by a tree of unique Newton polygons.
Then f has a unique Newton polygon at w and ] and assume that f has no unique Newton polygon at w, i.e., that there exists a vertex (i, trop(f i )(w)) ∈ ∆ w (f ) such that in w (f i ) is no monomial.Let µ 0 and µ 1 be the negated slopes of the edges after and before the vertex respectively, see Figure 3.
Then, for any w k ∈ (µ 0 , µ 1 ), we have in Next, we show the equality.For the "⊇" inclusion, let µ be a slope of an edge of ∆ w (f ), say connecting the two vertices v 0 and v 1 .Then, writing e(v 0 , v 1 ) for the edge connecting v 0 and v 1 , For the converse inclusion, let (w, w k ) ∈ Trop(f ).It is clear, that for some bounded proper face e ≤ ∆ w (f ), Note that e cannot be zero-dimensional, as otherwise in (w,w k ) (f ) = in (w,w k ) (f ) for all w k ∈ R, contradicting the zero-dimensionality of Trop(f ).Hence, e has to be an edge and, consequently, w k is the slope of e.

Proposition 2.16
For a triangular set F = {f 1 , . . ., f n } ⊆ K[x] the following are equivalent: (1) Moreover, if F is a tropical basis, then it admits a tree of unique Newton polygons.
Proof.We first show that (1) implies that F is a tropical basis and that it admits a tree of unique Newton polygons.By definition, we have Trop(f 1 ) = w 1 ∈Λ(f 1 ) {w 1 }× R n−1 .Applying Lemma 2.15 repeatedly, we see that for w 1 ∈ Λ(f 1 ), the polynomial f 2 has a unique Newton polygon at w 1 with and, for w 1 ∈ Λ(f 1 ) and w 2 ∈ Λ w 1 (f 2 ), f 3 has a unique Newton polygon at (w 1 , w 2 ) with and so forth.This shows on the one hand that F admits a tree of unique Newton polygons and on the other hand that any point in n i=1 Trop(f i ) corresponds to the component-wise valuation of a point in V (F ), implying that F is a tropical basis.
It remains to show that if (1) is not true, then F is no tropical basis.Assume for the sake of simplicity that dim Trop(f 1 ) ∩ Trop(f 2 ) = n − 1.Because Trop(f 1 ) = w 1 ∈Λ(f 1 ) {w 1 }×R n−1 and Trop(f 2 ) is invariant under translation by {(0, 0)}×R n−2 , there necessarily exist Trop(f i ), and since n i=1 Trop(f i ) is not zero-dimensional, F cannot be a tropical basis of the zero-dimensional ideal it generates.
From Proposition 2.16, we conclude that a generic triangular set is a tropical basis and admits a tree of unique Newton polygons in the following sense: Corollary 2.17 Let (K * ) N ⊆ K[x] n be the coefficient space of all triangular sets with fixed support.Then, in the topology induced by the valuation, there exists an open dense set U ⊆ (K * ) N such that any triangular set F ∈ U is a tropical basis and admits a tree of unique Newton polygons.
Proof.Consider the component-wise valuation ν : (K * ) N → R N .There exists an euclidean open dense subset U ⊆ R n such that the tropical hypersurfaces of any triangular set F ∈ (K * ) N with ν(F ) ∈ U intersect transversally as in Proposition 2.16 (1).As ν is continuous, its preimage U := ν −1 U ⊆ (K * ) N is also open and dense.

Computing tropical starting points
In this section, we use Algorithm 2.10 to compute points on higher-dimensional tropical varieties.This is done by reducing the dimension to zero by intersecting with randomly chosen hyperplanes.Moreover, we will use the algorithm to sample random maximal Gröbner cones on the tropical Grassmannians G 3,7 , G 4,7 , G 3,8 , G 4,8 and show that the latter two are not simplicial.
Proposition 3.1 Let I K[x] be a prime ideal of dimension d and X = V (I) its corresponding irreducible affine variety such that X ∩ (K * ) n = ∅.W.l.o.g.let {x 1 , . . ., x d } be algebraically independent modulo I. Then there exists a non-empty, Zariski open subset Proof.Abbreviating H λ := V ( x i − λ i | i = 1, . . ., d ), it is clear that there exists a Zariski open U 0 ⊆ (K * ) d with ∅ = X ∩ H λ and dim(X ∩ H λ ) = 0. Now consider the set in which the inclusion does not hold.It naturally decomposes into n − d subsets: As U can be chosen to be U 0 \ A, where (•) denotes the Zariski closure in (K * ) d , it suffices to show that A i = (K * ) d .This is easy to see: Because X is irreducible and X ∩ (K * ) n = ∅, we necessarily have dim(X ∩ V (x i )) < d for all i = d + 1, . . ., n.In particular, dim π(X ∩ V (x i )) < d, where π : K n K d is the canonical projection onto the first d coordinates.And, by construction, A i ⊆ π(X ∩ V (x i )).Proposition 3.1 can be reformulated into the following algorithm.Set I z to be the image of I under the substitution map x i else. .However, this requires one transcendental extension of K per hyperplane, which is not feasible in high codimension.
We will briefly define the examples of our interest.
) is defined to be the tropicalization of the ideal Grass(k, n) K[p], where the variables of the ring minors of any k × n matrix and the ideal Grass(k, n) is generated by all Plücker relations amongst them, see [MS15, Section 4.3].We consider the variables of K[p] to be sorted lexicographically, i.e.
Moreover, we define the ideal Det(k, n) K[x 11 , x 12 , . . ., x nn ] to be the ideal generated by the k × k minors of the matrix (x ij ) i,j=1,...,n .
In addition to computing starting points for the tropical traversals, Algorithm 3.2 can be used to sample random points on tropical varieties.
Example 3.6 (G k,n for k ∈ {3, 4} and n ∈ {7, 8}) Using Algorithm 3.2, we sampled random maximal cones on higher tropical Grassmannians ignoring symmetry.This was done by computing Gröbner cones around random tropical points, dismissing those of lower dimension and duplicates.We analyzed over 1000 distinct maximal cones on each of G 3,7 , G 4,7 and G 3,8 , as well as over 100 distinct maximal cones on the tropical variety of G 4,8 .
All cones were invariant under tensoring with F 2 , which is not surprising for G 3,7 : Even though Speyer and Sturmfels showed that G 3,7 depends on the characteristic of the ground field, in fact it is the smallest tropical Grassmannian depicting this behavior [SS04, Theorem 3.7], Herrmann, Jensen, Joswig and Sturmfels showed that, out of the 252 000 maximal cones of G 3,7 , this is only visible on a single cone, the Fano cone [HJJS09, Theorem 2.1].
Of the 1000 Gröbner cones sampled from each of G 3,7 and G 4,7 , every single one was simplicial, which was expected as G 3,7 is known to be simplicial [HJJS09, Theorem 2.1] and G 4,7 = G 3,7 by duality.In the 1000 and 100 Gröbner cones sampled from G 3,8 and G 4,8 respectively, each contained exactly one cone which was not simplicial, see the proof of Theorem 3.7.
As an immediate result, we obtain: Theorem 3.7 The tropical Grassmannian G d,n is not simplicial for d = 3, 4 and n = 8.
We conclude the section with some timings.testing all rays for containment in the tropical variety.It can also be found in Singular, however that implementation is slower than gfan.Singular 4.1.0:Algorithm 3.2, as implemented in tropicalNewton.lib.As gfan additionally computes a corresponding reduced Gröbner basis, we also provide analogous timings in Singular.
We would like to stress that these timings merely serve as a comparison of the algorithms and not as a showcase of the computational reach of the two systems involved.For instance, points on tropical Grassmannians can also be computed via the tropical Stiefel map, see [HJS14, Proposition 12] and [FR15].In fact, G 3,7 has been previously computed using gfan 0.
x i + q k x 2 0 , where q k ∈ N is the k-th prime.
While substituting x 0 → t directly yields triangular sets, we described in Example 2.12 how our Algorithm 2.10 struggles with them: It requires a degree 2 n field extension of Q, which results in a runtime of 8 seconds for n = 13, roughly doubling with each increase of n.However, for [BJSST07, Algorithm 9], this family is completely trivial: As F is already a reduced Gröbner basis for a suitable ordering, it is easy to verify that its ideal is one-dimensional and has a one-dimensional homogeneity space generated by (1, 1, . . ., 1) ∈ R n+1 .Hence [BJSST07, Algorithm 9] immediately obtains its tropical variety, which is equal to its homogeneity space.This shows in the runtime of both gfan 0.5 and gfan 0.6.2, which terminate instantaneously for n = 13 and whose runtimes remain under 1 second for n < 120.
x i else.
3: for i = d, . . ., n do 4: Let J ± i be images of J under the maps x i → p ±1 respectively.

7:
Set Scale elements of W ± i positively so that they are primitive in Z n .9: return W := computing Newton-Puiseux expansions Experiments suggest that, in both algorithms, the latter two bottlenecks are timewise insignificant compared to the first.In fact, for Algorithm 4.6, constructing the triangular decomposition from a lexicographical Gröbner basis is polynomial [Laz92, Section 7], as is the construction of the Newton-Puiseux expansion [Chi86].Hence, the main bottleneck in both algorithms lies in the computation of Gröbner bases with respect to elimination orderings.
However, the key difference is that these Gröbner basis computations in [Cha13, Algorithm 4.2.5]involve the one-dimensional input ideal I, whereas the ideals J ± i in Algorithm 4.6 are all zero-dimensional.For these ideals we not only have better complexity bounds [Laz83], but also techniques such as fglm [FGLM93], which speed up our calculations drastically.For instance, in the following Example 4.8 and in Singular 4.1.0,a lexicographical Gröbner basis of J ± i required only 30 seconds of computation while an elimination ideal of I required 25 minutes.
The reduced Gröbner basis of the initial ideal under w with respect to the reverse lexicographical ordering consists of 5543 binomials with degrees ranging from 2 to 7. The Gröbner cone C w (I) is simplicial with its 12 facets.Figure 6 shows some data on the reduced Gröbner bases of the saturated initial ideals under weight vectors on the facets of C w (I).The rows represent binomials, trinomials and quadrinomials respectively and the columns represent degrees 2 to 7, i.e. the entry in row i and column j is the number of Gröbner basis elements with i + 1 monomials and of degree j + 1.The computation of the 12 tropical links using Algorithm 4.6 took 7 minutes, while all attempts to compute any of the 12 tropical prevarieties failed to terminate within an hour, even using the newly developed techniques by Jensen, Sommars and Algorithm 3.2 (tropical point) Input: I K[x] prime ideal with V (I) ∩ (K * ) n = ∅.Output: w ∈ Trop(I).1: Compute a maximal algebraically indep.set modulo I, say {x 1 , . . ., x d }. 2: repeat 3: Pick z = (z 1 , . . ., z d ) ∈ (K * ) d randomly.

5:
until dim(I z ) = 0 and V (I z ) ⊆ (K * ) n−d 6: Compute a triangular set F ⊆ K[x d+1 , . . ., x n ] with √ I z ⊆ F .7: Compute a point (w d+1 , . . ., w n ) ∈ Trop(F ) using Algorithm 2.10.8: return (ν(z), w d+1 , . . ., w n ) Remark 3.3 (1) Randomized algorithms such as Algorithm 3.2 are commonly referred to as Las Vegas algorithms.This means that its result is always correct, however it only has an expected finite runtime.Nevertheless, Proposition 3.1 shows that generic choices of z in Step 3 will lead to termination.(2) Note that the set of all w ∈ R d such that {w} × R n−d does not intersect any lower-dimensional Gröbner polyhedra on Trop(I) is open and dense in the euclidean topology.Hence generic choices of z ∈ (K * ) d in Step 3 will also guarantee that the resulting tropical point will lie in the relative interior of a maximal Gröbner polyhedra on the tropical variety.(3) It is possible to eliminate the randomness by computing stable intersections with affine hyperplanes, as in a recent work of Jensen and Yu [JY16]

Timings 3. 8
Figure 4 compares three different algorithms for computing points on tropical varieties: gfan 0.6.2:an experimental algorithm based on Chan's work on tropical curves [Cha13, Chapter 4], and Jensen and Yu's work on stable intersections [JY16].gfan 0.5: [BJSST07, Algorithm 9], a random traversal of the Gröbner fan while

Figure 6 .
Figure 6.reduced Gröbner bases around a maximal cone in G 4,9 terms of lower x k -degree for some c k ∈ K * and d k ∈ N >0 .