Related papers: The tropical double description method
This paper investigates the geometric properties of a special case of the two-sided system given by $2 \times 2$ tropical commuting constraints. Given a finite matrix $A \in \mathbb{R}^{2\times 2}$, the paper studies the extremals of the…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact…
We study singularities in tropical hypersurfaces defined by a valuation over a field of positive characteristic. We provide a method to compute the set of singular points of a tropical hypersurface in positive characteristic and the p-adic…
The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: Where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying…
The problem of whether and how one can compute the twin-width of a graph -- along with an accompanying contraction sequence -- lies at the forefront of the area of algorithmic model theory. While significant effort has been aimed at…
To quantify the geometric expressivity of transformers, we introduce a tropical geometry framework to characterize their exact spatial partitioning capabilities. By modeling self-attention as a vector-valued tropical rational map, we prove…
We introduce the first graph kernels for metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels based on graph combinatorics such as nodes, edges, and subgraphs, our metric graph kernels are purely based on…
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…
We give several characterizations of stable intersections of tropical cycles and establish their fundamental properties. We prove that the stable intersection of two tropical varieties is the tropicalization of the intersection of the…
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…
We propose an algebraic geometric framework to study the expressivity of linear activation neural networks. A particular quantity of neural networks that has been actively studied is the number of linear regions, which gives a…
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…
We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals…
Enumerating the extremal submodular functions defined on subsets of a fixed base set has only been done for base sets up to five elements. This paper reports the results of attempting to generate all such functions on a six-element base…
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…
We introduce some analytic relations on the set of partial differential equations of two variables. It relies on a new comparison method to give rough asymptotic estimates for solutions which obey different partial differential equations.…
We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems…