Minimal Representations of Tropical Rational Functions

TL;DR

This paper investigates minimal algebraic representations of tropical rational functions, establishing uniqueness in certain cases and providing bounds related to tropical hypersurface arrangements and Minkowski sums.

Contribution

It introduces two notions of minimality for tropical rational functions, analyzes their properties across dimensions, and extends existing bounds on tropical hypersurface arrangements and Minkowski sums.

Findings

In dimension one, both minimality notions coincide and representations are unique.

Provides counting formulas and lower bounds for tropical hypersurface arrangements.

Extends bounds related to Minkowski sums and tropical hypersurfaces.

Abstract

This paper studies the following question: given a piecewise-linear function, find its minimal algebraic representation as a tropical rational signomial. We put forward two different notions of minimality, one based on monomial length, the other based on factorization length. We show that in dimension one, both notions coincide, but this is not true in dimensions two or more. We prove uniqueness of the minimal representation for dimension one and certain subclasses of piecewise-linear functions in dimension two. As a proof step, we obtain counting formulas and lower bounds for the number of regions in an arrangement of tropical hypersurfaces, giving a small extension for a result by Mont\'{u}far, Ren and Zhang. As an equivalent formulation, it gives a lower bound on the number of vertices in a regular mixed subdivision of a Minkowski sum, giving a small extension for Adiprasito's Lower Bound Theorem for Minkowski sums.