Symmetric and r-symmetric tropical polynomials and rational functions

A tropical polynomial in nr variables, divided into blocks of r variables each, is r  -symmetric if it is invariant under the action of SnSn that permutes the blocks. For r=1r=1 we call these symmetric tropical polynomials. We can define r-symmetric and symmetric tropical 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 for r≥2r≥2, we show that r-symmetric tropical rational functions are and provide a list of generators.