Related papers: Tropical function fields, finite generation, and f…
We prove an analogue of Kirchhoff's matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym…
We show how tropical varieties of ideals I over a field K with non-trivial valuation can be traced back to tropical varieties of ideals in R[[t]][x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we…
We consider functions $f:B\to\Rset$ that obey tropical analogs of classical Pl\"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form $\{x\in \Zset^n\colon 0\le x\le a, m\le…
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…
We introduce the tropicalization of closed subschemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the results of Einslieder, Kapranov, Lind,…
We construct the scaling site S by implementing the extension of scalars on the arithmetic site, from the smallest Boolean semifield to the tropical semifield of positive real numbers. The obtained semiringed topos is the Grothendieck topos…
Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs…
A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic…
We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…
We introduce a relative refined $\chi_y$-genus for sch\"on subvarieties of algebraic tori. These are rational functions of degree minus the codimension with coefficients in the ring of lattice polytopes. We prove that the relative refined…
We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical…
I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a…
We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the…
There is a well known correspondence between the triangle inequality for a distance function on a finite set, and idempotency of an associated matrix over the tropical semiring. Recent research has shed new light on the structure…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. To achieve this, we develop…
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…
We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements…
Let $X$ be a smooth geometrically connected projective curve of genus two over a complete non-archimedean field $K$. For discretely valued $K$, the first main theorem in \cite{liu} gives a set of criteria on the Igusa invariants of the…
The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of…