Related papers: Combinatorial and inductive methods for the tropic…
We give combinatorial proofs of several recent results due to Merca on the sum of different parts congruent to $r$ modulo $m$ in all partitions of $n$. The proofs make use of some well known involutions from the literature and some new…
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…
We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine…
We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We…
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…
We prove a conjecture of W.~Hackbusch in a bigger generality than in our previous article. Here we consider Tensor Train (TT) model with an arbitrary number of leaves and a corresponding "almost binary tree" for Hierarchical Tucker (HT)…
We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise…
We develop estimation for potentially high-dimensional additive structural equation models. A key component of our approach is to decouple order search among the variables from feature or edge selection in a directed acyclic graph encoding…
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…
Consider a collection of m competing machine learning algorithms. Given their performance on a benchmark of datasets, we would like to identify the best performing algorithm. Specifically, which algorithm is most likely to ``win'' (rank…
Via correspondence theorems, rational log Gromov--Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalized)…
Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…
The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications…
We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $\RP^n$ is maximal. That is, there exist generic configurations of real linear spaces such…
We develop a tropical analog of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in O(n(m+n)) time, where m…
We introduce new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from the appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index)…
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
In this work, we examine the process of Tropical Polynomial Division, a geometric method which seeks to emulate the division of regular polynomials, when applied to those of the max-plus semiring. This is done via the approximation of the…
We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we…