Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

Abstract

Let $g_1,\dots,g_k$ be tropical polynomials in n variables with Newton polytopes $P_1,\dots,\allowbreak P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1,\dots,g_k$, such as the f-vector, the number of unbounded faces, and (in the case of a curve) the genus. Our point of departure is Vigeland's work [Tropical complete intersection curves, preprint, arXiv:math/0711.1962, 2007] which 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 $P_1,\dots,P_k$. 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.