Every tropical hypersurface is the union of finitely many polyhedra. By the distributive law, intersecting a list of tropical hypersurfaces amounts to doing a huge computation of intersections of polyhedra that may be naturally organised in an enumeration tree. The idea of dynamic enumeration proposed by Mizutani, Takeda and Kojima (2007) is extended to dynamic decomposition where tropical hypersurfaces are not just chosen dynamically but also split dynamically into disjoint halfopen polyhedra. This has the advantage of making the enumeration tree even thinner. We present an implementation of this enumeration algorithm in the C++ library gfanlib and do a comparison against an implementation of an existing algorithm. Finally we give an example of how the software has been used in a celestial mechanics problem.

Inspired by numerical homotopy methods we propose a combinatorial homotopy algorithm for finding all isolated solutions to a tropical polynomial systems of n tropical polynomials in n variables. In particular, a tropicalisation of the numerical "regeneration" technique leads to a new method for enumerating the mixed cells of a mixed subdivision. This tropical approach shares some ideas with the recent algorithm by Malajovich. However, our algorithm has several advantages. It is memoryless, parallelisable as a tree traversal, exact and relies on symbolic perturbations. Our computational experiments show that the method is competitive and especially fast on the Katsura class of examples.

Mixed volumes of lattice polytopes play a central role in numerical and tropical algebraic geometry. We present an implementation of a new algorithm for their computation based on tropical homotopy continuation, which is a combinatorial procedure using ideas from numerical algebraic geometry. While the mathematical aspects of the algorithm are presented elsewhere, here we mainly address technical details of the implementation, in particular how it was made fast and reliable. The implementation is distributed as part of the library gfanlib.

We prove that a fixed configuration of N - 1 masses in the plane can be extended to a central configuration of N masses by adding a specified additional mass only in finitely many ways. This holds for a family of potential functions including the Newtonian gravitational case and the classical planar point vortex model.

Balas introduced intersection cuts for mixed integer linear sets. Intersection cuts are given by closed form formulas and form an important class of cuts for solving mixed integer linear programs. In this paper we introduce an extension of intersection cuts to mixed integer conic quadratic sets. We identify the formula for the conic quadratic intersection cut by formulating a system of polynomial equations with additional variables that are satisfied by points on a certain piece of the boundary defined by the intersection cut. Using a software package from algebraic geometry we then eliminate variables from the system and get a formula for the intersection cut in dimension three. This formula is finally generalized and proved for any dimension. The intersection cut we present generalizes a conic quadratic cut introduced by Modaresi, Kilinc and Vielma.

We propose an algorithm for computing the facets of a pure connected polyhedral fan up to symmetry. The fan is represented by an oracle. With suitable implementations of the oracle the same algorithm can be used for computing secondary fans, Gröbner fans, tropical varieties and Minkowski sums up to symmetry. The algorithm has been implemented in the software Gfan.

The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseuxvalued "lift" of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the "lifting algorithm" usingSingular and Gfan if the base field is ℚ. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K n+1, 0).

The Grobner fan of an ideal in the polynomial ring in n variables is an n-dimensional polyhedral complex and the tropical variety of the ideal is a certain subcomplex. In this paper we describe the software Gfan for computing these fans. Computing the Grobner fan is equivalent to computing all the reduced Grobner bases of the ideal.

Gfan is a software package for computing Grobner fans and tropical varieties. These are polyhedral fans associated to polynomial ideals. The maximal cones of a Grobner fan are in bijection with the marked reduced Grobner bases of its defining ideal. The software computes all marked reduced Grobner bases of an ideal. Their union is a universal Grobner basis. The tropical variety of a polynomial ideal is a certain subcomplex of the Grobner fan. Gfan contains algorithms for computing this complex for general ideals and specialized algorithms for tropical curves, tropical hyper- surfaces and tropical varieties of prime ideals. In addition to the above core functions the package contains many tools which are useful in the study of Grobner bases, initial ideals and tropical geometry. Among these are an interactive traversal program for Grobner fans and programs for graph- ical renderings. The full list of commands can be found in Appendix B. For ordinary Grobner basis computations Gfan is not competitive in speed compared to programs such as CoCoA, Singular and Macaulay2.