相关论文: Computing Tropical Varieties
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
We outline a generalization of the Groebner fan of a homogeneous ideal with maximal cells parametrizing truncated Groebner bases. This "truncated" Groebner fan is usually much smaller than the full Groebner fan and offers the natural…
In this note I will explain how relative/log Gromov-Witten invariants of pairs $(X,D)$ with very ample smooth anticanonical divisor $D$ can be computed using algebro-combinatorial objects called scattering diagrams. The underlying principle…
In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…
We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge…
We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in $\mathbb{TP}^n$. We provide a lower estimate for the maximal value of the top Betti number, which naturally depends on the dimension…
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman…
This is the first part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. The main result of this part is a constructibility result for the analytic Gr\"obner fan of a…
The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of…
Let K be a field with a valuation and let S be the polynomial ring S:= K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in…
This paper shows the polytopality of any finite type $\mathbf{g}$-vector fan, acyclic or not. In fact, for any finite Dynkin type $\Gamma$, we construct a universal associahedron $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$ with the property that…
The GIT chamber decomposition arising from a subtorus action on a quasiprojective toric variety is a polyhedral complex. Denote by Sigma the fan that is the cone over the polyhedral complex. In this paper we show that the toric variety…
We present simple compact proofs of the strong and weak duality theorems of tropical linear programming. It follows that there is no duality gap for a pair of tropical primal-dual problems. This result together with known properties of…
In a previous work, we stated the conjecture that any semiring homomorphism between the Boolean function semirings on 1-dimensional tropical fans has the property called geometric. In this paper, we show that the conjecture is true. As an…
This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as…
We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…
Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant…
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling…
The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel-Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical…