Related papers: Euler Characteristic of real nondegenerate tropica…

In analogy to chapter 9 of arXiv:0709.3705 we define an intersection product of tropical cycles on tropical linear spaces L^n_k, i.e. on tropical fans of the type max{0,x_1,...,x_n}^(n-k)*R^n. Afterwards we use this result to obtain an…

In 2022, Brugall{\'e} and Jaramillo-Puentes showed that the coefficients of small codegree of the tropical refined invariant are polynomial in the Newton polygon. This raised the question of the existence of universal polynomials giving…

The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to pass to a suitable compactification, e.g., by one point. This observation can be exploited to prove that any tropical…

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…

D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a…

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical…

We introduce a new non-degeneracy condition at infinity for a real or a mixed polynomial mapping $F$ which allows us to approximate its bifurcation locus in terms of certain Newton polyhedra. We derive a sufficiency result for the Jacobian…

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…

The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of…

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials…

We characterise Newton polytopes of nondegenerate quadratic forms and Newton polyhedra of Morse singularities.

Using the interpretation of the ultradiscretization procedure as a non-Archimedean valuation, we use results of tropical geometry to show how roots and poles manifest themselves in piece-wise linear systems as points of…

We generalize two classical formulas for complete intersection curves by introducing the the complete intersection discrepancy of a curve as a correction term. The first is a well-known multiplicity formula in singularity theory, due to…

We define the characteristic cycle of a constructible sheaf on a smooth surface in the cotangent bundle. We prove that the intersection number with the 0-section equals the Euler number and that the total dimension of vanishing cycles at an…

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We…

A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes $\Delta_s(\mathcal{G})$ of multigraphs $\mathcal{G}$, which…

For a tropical univariate polynomial $f$ we define its tropical Hilbert function as the dimension of a tropical linear prevariety of solutions of the tropical Macauley matrix of the polynomial up to a (growing) degree. We show that the…

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We…

The work is a detailed study of rational singularities of multiplicity 3 (RTP-singularities, for short). We give a list of nonisolated hypersurface singularities of which normalisations are the RTP-singularities, and construct their minimal…

In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…