Symmetric Polynomials in Upper-bound Semirings

TL;DR

This paper characterizes when upper-bound semirings allow symmetric polynomials to be expressed via elementary symmetric polynomials, extending classical results from rings to a broader semiring context.

Contribution

It provides a complete characterization of fully elementary upper-bound semirings and refines this for linearly ordered cases, expanding the understanding of symmetric polynomials in semirings.

Findings

Complete characterization of fully elementary upper-bound semirings

Extension of classical symmetric polynomial results to semirings

Improved results for linearly ordered upper-bound semirings

Abstract

The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The result does not extend directly to polynomials over semirings, but we do have analogous results for some special semirings, for example, the tropical, extended and supertropical semirings. These all fall into a larger class of upper-bound semirings. In this paper we extend the known results and give a complete characterization of fully elementary upper-bound semirings. We further improve this characterization statement in the case of linearly ordered upper-bound semirings.