Related papers: Tropical conics for the layman

In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical…

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em…

We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly…

The problem of solving tropical linear systems, a natural problem of tropical mathematics, has already proven to be very interesting from the algorithmic point of view: it is known to be in $NP\cap coNP$ but no polynomial time algorithm is…

Reduction trees are a way of encoding a substitution procedure dictated by the relations of an algebra. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We introduce and study minimal (with respect to inclusion) solutions of systems of tropical linear differential equations. We describe the set of all minimal solutions for a single equation. It is shown that any tropical linear differential…

In a series of articles, we have been developing a theory of tropical diagrams of probability spaces, expecting it to be useful for information optimization problems in information theory and artificial intelligence. In this article, we…

We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We…

Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system…

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical…

We define the tropical Tevelev degrees, $\mathsf{Tev}_g^{trop}$, as the degree of a natural finite morphism between certain tropical moduli spaces, in analogy to the algebraic case. We develop an explicit combinatorial construction that…

This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…

In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…

The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the…

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…

This paper, a continuation of [3], involves a closer study of polynomials of supertropical semirings and their version of tropical geometry in which we introduce the concept of relatively prime polynomials and resultants, with the aid of…

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric…