On Symmetric Polynomials

TL;DR

This paper explores the structure of symmetric function algebras, introducing perturbation-based generators and algebra-monomorphisms, and provides a JAVA algorithm for computations involving elementary symmetric polynomials.

Contribution

It presents a novel approach to understanding symmetric polynomial algebras through perturbations and develops an algorithm for related computations.

Findings

Established inductive structure theorems for symmetric polynomial algebras

Defined perturbation-based generators and algebra-monomorphisms

Provided a JAVA algorithm for computations with elementary symmetric polynomials

Abstract

In this paper, we study structure theorems of algebras of symmetric functions. Based on a certain relation on elementary symmetric polynomials generating such algebras, we consider perturbation in the algebras. In particular, we understand generators of the algebras as perturbations. From such perturbations, define injective maps on generators, which induce algebra-monomorphisms (or embeddings) on the algebras. They provide inductive structure theorems on algebras of symmetric polynomials. As application, we give a computer algorithm, written in JAVA v. 8, for finding quantities from elementary symmetric polynomials.