Symmetric Polynomials in Tropical Algebra Semirings

TL;DR

This paper investigates symmetric polynomials within various tropical algebra semirings, identifying conditions under which they can be expressed via elementary symmetric polynomials, thus extending the classical fundamental theorem to these semirings.

Contribution

It characterizes the semirings where symmetric polynomials can be expressed through elementary symmetric polynomials, extending the fundamental theorem of symmetric polynomials to tropical algebra.

Findings

Identifies semirings where symmetric polynomials are expressible via elementary ones

Determines the extent of the fundamental theorem's applicability in tropical semirings

Provides a framework for understanding symmetric polynomials in extended tropical algebra contexts

Abstract

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +), Izhakian-Rowen's extended and supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. This allows us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.