Symmetric and r-Symmetric Tropical Polynomials and Rational Functions

TL;DR

This paper investigates r-symmetric tropical polynomials and rational functions, identifying generators for the latter and demonstrating their finite generation, unlike the former.

Contribution

It establishes that r-symmetric tropical rational functions are finitely generated and provides explicit generators, contrasting with the non-finite generation of r-symmetric tropical polynomials.

Findings

r-symmetric tropical polynomials are not finitely generated

r-symmetric tropical rational functions are finitely generated

explicit generators for r-symmetric rational functions are provided

Abstract

A tropical polynomial in nr variables divided into blocks of r variables each, is r-symmetric, if it is invariant under the action of Sn that permutes the blocks. For r=1 we call these tropical polynomials symmetric. We can define r-symmetric and symmetric rational functions in a similar manner. In this paper we identify generators for the sets of symmetric tropical polynomials and rational functions. While r-symmetric tropical polynomials are not finitely generated, we show that r-symmetric rational functions are and provide a list of generators.