Minimum volumes of tropical rational functions

TL;DR

This paper introduces a duality theorem for tropical rational functions, defining a volume measure for pairs of tropical polynomials, and explores minimal volume representations in one and two dimensions.

Contribution

It develops a duality framework for tropical rational functions and characterizes minimal volume representations, highlighting differences between one and two dimensions.

Findings

Minimal volume representations exist for n=1.

Dual subdivision is unique up to translation in one dimension.

Uniqueness of dual subdivision fails in two dimensions.

Abstract

When a tropical rational function \varphi on R^n is given, we can represent it as \varphi=f-g with tropical polynomials f and g. We develop the duality theorem for tropical rational functions to define the volume of the pair (f, g). We show that when n=1, we can find a representation of \varphi(x) \neq -\infty as f(x)-g(x) with the pair (f, g) of minimum volume. The dual subdivision of f(x) \oplus (yg(x)) is unique up to translation, but when n=2 this is not true.