Related papers: Computing tropical points and tropical links
Decomposing an algebraic variety into irreducible or equidimensional components is a fundamental task in classical algebraic geometry and has various applications in modern geometry engineering. Several researchers studied the problem and…
A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean. We provide a solution for a class of semidefinite feasibility…
Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to…
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…
This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are…
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…
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral…
We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen…
Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with…
While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural…
The decomposition of polynomial spaces on unions of Grassmannians $\mathcal G_{{k_1},d}\cup\ldots\cup \mathcal G_{{k_r},d}$ into irreducible orthogonally invariant subspaces and their reproducing kernels are investigated. We also generalize…
We investigate uniqueness issues that arise in $l^\infty$-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of $\mathbb{R}^n$ based on the dimension of the set of…
We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We…
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…
This paper provides a rigorous study of tropicalizations of locally symmetric varieties. We give applications beyond tropical geometry, to the cohomology of moduli spaces as well as to the cohomology of arithmetic groups. We study two cases…
We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking.…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…