Related papers: Tropical Laurent series, their tropical roots, and…

We give a general criterion for two toric varieties to appear as fibers in a flat family over the projective line. We apply this to show that certain birational transformations mapping a Laurent polynomial to another Laurent polynomial…

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero,…

Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible…

We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…

Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In…

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 consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…

Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical…

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…

We give a description of the tropical variety of univariate polynomials of degree n having two double roots. As a set, it is given as the union of three types of maximal cones of dimension n-1, where only cones of two of these types are…

We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.

Tropicalization is a procedure that assigns polyhedral complexes to algebraic subvarieties of a torus. If one fixes a weighted polyhedral complex, one may study the set of all subvarieties of a toric variety that have that complex as their…

The main mathematical focus of this paper is a class of parametrised polynomial systems that we refer to as being tropically transverse. We show how their generic number of solutions can be expressed as the mixed volume of a modified…

We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining "toric" roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more…

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is…

In a recent paper, Matthew Baker and Oliver Lorscheid showed that Descartes's Rule of Signs and Newton's Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields…

We consider the action of a permutation group $G$ of order $k$ on the tropical polynomial semiring in $n$ variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if $G$ is generated by…

We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our…

The tropical Stiefel map associates to a tropical matrix A its tropical Pluecker vector of maximal minors, and thus a tropical linear space L(A). We call the L(A)s obtained in this way Stiefel tropical linear spaces. We prove that they are…

Let $f(x)$ be a monic polynomial over $\mathbb{Q}$ with complex roots $\alpha_1,\dots,\alpha_n$. Linear relations among them and $1$ over $\mathbb{Q}$ play an important role when we study the distribution of roots modulo a prime. We study…