Related papers: Generic root counts and flatness in tropical geome…
A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
In this paper, we review the eigenpair problem in the context of tropical algebra. An important fact that has been largely overlooked in spectral theory of tropical algebra is that the tropical algebraic eigenvalues, which are obtained from…
In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed…
In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the…
We study tropical geometry in the global setting using Berkovich's deformation retraction. We state and prove the generalized balancing conditions in this setting. Starting with a strictly semi-stable formal scheme, we calculate certain…
We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…
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…
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of…
Counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$ with fixed contact order with the toric boundary and satisfying point conditions can be determined with tropical methods by Mikhalkin. If we require that our curves intersect the zero-…
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.…
Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality…
This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They…
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…
A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure…
In this study we extend the concepts of $m$-pluripotential theory to the Riemannian superspace formalism. Since in this setting positive supercurrents and tropical varieties are closely related, we try to understand the relative capacity…
Let X and X' be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles, in the sense of Henning Meyer's graduate thesis, between the tropicalization of the intersection…