Related papers: A tropical version of Hilbert polynomial (in dimen…
We define the unipotent tropical fundamental group of a polyhedral complex in $\mathbb{R}^n$ as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable…
Steingrimsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley-Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the…
We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an…
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +),…
We find a relation between mixed volumes of several polytopes and the convex hull of their union, deducing it from the following fact: the mixed volume of a collection of polytopes only depends on the product of their support functions…
We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…
The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When…
We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$,…
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties…
We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining…
Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state…
We study the combinatorial properties of 2-dimensional tropical complexes. In particular, we prove tropical analogues of the Hodge index theorem and Noether's formula. In addition, we introduce algebraic equivalence for divisors on tropical…
We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the…
When a tropical rational function \varphi on R^n is given, we can represent it as \varphi=f-g with tropical polynomials f and g. We develop the duality theorem for tropical rational functions to define the volume of the pair (f, g). We show…
Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we develop an…
In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces.…
This paper is a continuation of [arXiv:1603.02204]. Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove…
We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a…
In this article, we introduce an exponential for tropical matrices and show that this series is essential for the analysis of certain kinds of stability in discrete event dynamic systems. A notion of a generalised eigenvector is introduced…
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…