Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces

TL;DR

This paper establishes a correspondence between tropical theta functions and convex polyhedra, and proves a Riemann-Roch inequality for tropical abelian surfaces through divisor intersection calculations.

Contribution

It introduces a novel geometric interpretation of tropical theta functions and extends the Riemann-Roch inequality to tropical abelian surfaces.

Findings

Theta functions on tropical tori form a convex polyhedron

Riemann-Roch inequality holds for tropical abelian surfaces

Self-intersection numbers of divisors are explicitly calculated

Abstract

We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.