[2205.05647] minimal representation of tropical rational functions

Download the PDF of the paper entitled “Minimum Representations of Tropical Rational Functions” by Ngoc Mai Tran and one other author.

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overview: This paper studies the following questions: Given a piecewise linear function, find its minimal algebraic representation as a tropical rational symbol. One based on the length of the monomial and one based on the length of the factorization. We show that both concepts are the same in one dimension, but this is not true in two or more dimensions. Prove the uniqueness of the minimal representation of a given subclass of piecewise linear functions of dimensions 1 and 2. As a proof step, we obtain formulas and lower bounds for the number of regions in the arrangement of tropical hypersurfaces, slightly extending the results by Montúfar, Ren, and Zhang. As an equivalent formulation, we give a lower bound on the number of vertices in regular mixed refinements of Minkowski sums, and give a small extension to the Adiplasit lower limit theorem for Minkowski sums.