Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

TL;DR

This paper explores the combinatorial and topological properties of tropical intersections, generalizing previous results to arbitrary cases and linking tropical geometry with Ehrhart theory and toric varieties.

Contribution

It extends Vigeland's work to general Newton polytopes and intersection dimensions, providing new formulas for face counts and genus using mixed volumes.

Findings

Derived formulas for face counts and genus in tropical intersections.

Established a connection between tropical and toric intersection genus.

Developed aspects of a mixed Ehrhart theory for these polytopes.

Abstract

Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in terms of mixed volumes. By establishing some aspects of a mixed version of Ehrhart theory we show that the genus of a tropical intersection curve equals the genus of a toric intersection curve corresponding to the same Newton polytopes.