Minimal Representations of Tropical Rational Functions

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.