Related papers: Lifting non-proper tropical intersections

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,…

Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover,…

The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete…

In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there…

The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical…

This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra,…

Let K be an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. Let X be a K-rigid analytic variety and \Sigma a semianalytic subset of X. Then the closure of \Sigma in X with respect to the…

We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the…

This paper constructs (with challenging obstacles) on the three torus with its cubical decomposition: Firstly, a combinatorial graded intersection algebra (graded by the codimension) which is commutative and associative defined by…

This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…

We prove that for every tower $\mathcal T$ there are $\aleph_1$-dense $A$ and $B$ so that any ``reasonable" forcing notion $\mathbb{P}$ -- an adjective that includes all known ones -- for making $A$ and $B$ isomorphic will add a…

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the…

Given a C$^*$-correspondence $X$, we give necessary and sufficient conditions for the tensor algebra $\mathcal T_X^+$ to be hyperrigid. In the case where $X$ is coming from a topological graph we obtain a complete characterization.

In this paper we describe the multiplier ideals and jumping numbers associated with an irreducible germ of quasi-ordinary hypersurface $(D, 0) \subset (\mathbb{C}^{d+1}, 0)$ by using a toroidal embedded resolution. The approach is motivated…

Let X be a coherent configuration associated with a transitive group G. In terms of the intersection numbers of X, a necessary condition for the point stabilizer of G to be a TI-subgroup, is established. Furthermore, under this condition, X…

For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Amp\`ere equations, taking place on the associated Berkovich space, and the…

We use tropical and non-archimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space $Y$. In particular, we are interested in the choices of parameters for which the generic root…

This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on…

We present an unobstructedness criterion for Lagrangian threefolds $L\subset X^A$ using the $H_1(L)$-class associated with the boundary of a pseudoholomorphic disk. As an application, let $X^A\to Q$ be a Lagrangian torus fibration whose…