The TropLi Tutorial Felipe Rinc´on For version 0.1 Abstract TropLi is a software tool for computing tropical linear spaces of matroids represented by some integer matrix A. It is based on the algorithm presented in the paper Computing Tropical Linear Spaces by Felipe Rinc´ on for computing the cyclic Bergman fan of a matroid. TropLi can also be used to compute some matroidal invariants of the matrix A, like its collection of bases, circuits, and its Tutte polynomial. An additional tool called TropLi disc is available for computing the tropicalization of the A-discriminantal variety and vertices of the Newton polytope of the A-discriminant.

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Introduction and Preliminaries

Given a complex matrix A ∈ Cm×n with columns a1 , a2 , . . . , an ∈ Cm , its associated matroid M (A) is the matroid on the ground set [n] := {1, 2, . . . , n} encoding the linear dependences in Cm among the columns of A, i.e., M (A) is the matroid whose circuits are the sets C ⊆ [n] for P which there is a minimal dependence among the the columns of A of the form i∈C λi ai = 0. The tropical linear space T (M ) of any matroid M over the ground set [n] is defined to be the set of vectors v ∈ Rn such that for any circuit C of M , the minimum min{vi | i ∈ C} is attained at least twice (i.e., there exist j, k ∈ C distinct such that vj = vk = min{vi | i ∈ C}). In the case M is the matroid associated to some complex matroid A ∈ Cm×n , T (M ) agrees with the tropicalization of the subspace rowspace(A) ⊆ Cn (using the trivial valuation on C). Note that if the matroid M has a loop then T (M ) is the empty set, and if M has a coloop i ∈ [n] then T (M ) = T (M \ i) × R · ei , so from now on we will assume that the matroid M has no loops and no coloops. The cyclic Bergman fan Φ(M ) of a matroid M is a polyhedral fan supported in the tropical linear space T (M ), which was introduced in [Rin11]. Its maximal cones are in bijection with regressive compatible pairs, and its rays are in bijection with flats of M which are either cyclic flats or singletons. The cyclic Bergman fan of M is amenable to computational purposes, and a fast algorithm to compute it was presented in [Rin11]. TropLi is a software tool for computing the cyclic Bergman fan Φ(M ) of a matroid M associated to some integer matrix A ∈ Zm×n of rank m without 1

any loops or coloops. It is based on a C++ implementation of the algorithm described in [Rin11] for computing Φ(M ). TropLi uses the C++ library LEDA [Gmb] in order to perform row operations on the matrix A with exact integer arithmetic.

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Using TropLi

In order to use TropLi in a UNIX system, open a terminal window and navigate to the directory where TropLi is located. Use the program by typing ./tropli followed by an integer matrix A, for example, 5 1 0 0 0 0

16 1 1 0 0 0 0 0 1 1 0

1 0 0 1 1

1 0 1 0 0

1 0 1 0 1

1 0 1 1 0

1 0 1 1 1

1 1 0 0 0

1 1 0 0 1

1 1 0 1 0

1 1 0 1 1

1 1 1 0 0

1 1 1 0 1

1 1 1 1 0

1 1 1 1 1

where the first line just tells TropLi the dimensions of the matrix A. Optionally, you can store the input matrix A in a text file (say mymatrix.txt) in the same directory, and then redirect the input to that file by typing ./tropli
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You can tell TropLi to compare the cyclic Bergman fan Φ(M ) with the Bergman fan B(M ) of M , which is the coarsest fan structure on the tropical linear space of the matroid M , by using the flag “-compare”. Running the command ./tropli
44 46

which indicates that the union of the maximal cones #44 and #46 of the cyclic Bergman is the maximal cone of the Bergman fan indexed by the partition  {0, 3, 5, 6, 7, 8, 11, 13, 14, 15} , {1} , {2, 10} , {4, 12} , {9} of the set {0, 1, . . . , 15}, i.e., the maximal cone of B(M ) whose affine span is the 5-dimensional subspace of R16 described by the equations x0 = x3 = x5 = x6 = x7 = x8 = x11 = x13 = x14 = x15 , x2 = x10 , x4 = x12 . For more details on this labeling of maximal cones of B(M ) by partitions, see Corollary 4.7 in [FS05]. You can also run the command ./tropli n): in this case it is faster to run TropLi on the smaller (n − m) × n matrix A⊥ (whose rowspace is the orthogonal complement to the rowspace of A) and use the flag “-dual”. See examples 4.4 and 4.5 in [Rin11]. Running the command ./tropli
will compute the sets of bases and non-bases of the matroid M , producing the additional output Number of bases = 3008 Number of non-bases = 1360 List of bases written in the file:

tropli bases.out

The file tropli bases.out will then contain a list of all subsets of columns of A which are a basis for Qm in lexicographic order, with the columns labeled by the numbers 0, 1, . . . , n − 1. You can similarly type ./tropli
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TropLi disc: dealing with A-discriminants

You can use the program named TropLi disc to compute tropicalizations of A-discriminantal varieties and vertices of Newton polytopes of A-discriminants, based on computations of tropical linear spaces. For information on how this is done we refer the reader to [DFS07, Rin11]. You can use the program TropLi disc by typing ./tropli disc

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followed by an m × n integer matrix A and a vector w ∈ Zn , for example, 4 1 0 0 0

12 1 1 0 0 0 1 1 0

1 0 0 2

1 0 1 1

1 0 2 0

0 1 0 0

0 1 1 0

0 1 0 1

0 1 2 0

0 1 1 1

0 1 0 2

2 -3 0 1 4 -5 -9 2 0 -3 4 6 where the first line just indicates the dimensions of the matrix A. Optionally, you can store all this input in a text file (say myinput.txt) in the same directory, and then redirect the input to that file by typing ./tropli disc
TropLi disc will now perform a ray shooting algorithm (as described in [DFS07, Rin11]) to compute the vertex u of the Newton polytope N P (∆A ) of the Adiscriminant ∆A which minimizes the dot product u · w. In the case where w is not generic and this minimum is attained at several vertices of N P (∆A ), it will compute a random vector v ∈ Zn and use a symbolic perturbation procedure to output the vertex u minimizing the dot product u · (w +  · v), for  > 0 sufficiently small (note that the vertex u produced in this way will also minimize the dot product u · w). Once it has computed the vertex u, it will produce the output Vertex matrix written in the file: A-degree of the A-discriminant: 6 6 8 8 Total time elapsed: 9 seconds

disc vertices.out

where the A-degree of the A-discriminant is the vector A · u, with u any point in N P (∆A ) (this vector does not depend on the choice of u). The coordinates of the vertex u can be found in the file “disc vertices.out”. 5

Remark 3.2. The union of the computed cones {σ + rowspace(A) | σ + rowspace(A) ⊆ Rn has codimension 1 } is the tropicalization T (∇A ) of the A-discriminantal variety ∇A (in the case it has codimension 1), but these cones do not intersect nicely, i.e., two of these cones might intersect in their interior. However, this description of T (∇A ) can be used to compute the normal fan of the Newton polytope of the Adiscriminant, as described in [JY11]. You can type, for example, ./tropli disc
References [DFS07] Alicia Dickenstein, Eva Maria Feichtner, and Bernd Sturmfels, Tropical discriminants, J. Amer. Math. Soc. 20 (2007), no. 4, 1111–1133. [FS05]

Eva Maria Feichtner and Bernd Sturmfels, Matroid polytopes, nested sets and Bergman fans, Port. Math. (N.S.) 62 (2005), no. 4, 437–468.

[Gmb]

Algorithmic Solutions Software GmbH, Leda free edition, Available at http://www.algorithmic-solutions.com/leda/ledak/index.htm.

[JY11]

Anders Jensen and Josephine Yu, Computing tropical resultants, arXiv:1109.2368.

[Rin11] Felipe Rinc´ on, Computing tropical linear spaces, arXiv:1109.4130.

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