Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Separating symmetric polynomials over finite fields

Artem Lopatin, Pedro Antonio Muniz Martins, Lael Viana Lima

Abstract

The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb F}_q$ the set $S(n)$ is not a minimal separating set for symmetric polynomials in general. We determined when $S(n)$ is a minimal separating set for the algebra of symmetric polynomials having the least possible number of elements.

Separating symmetric polynomials over finite fields

Abstract

The set of all elementary symmetric polynomials in variables is a minimal generating set for the algebra of symmetric polynomials in variables, but over a finite field the set is not a minimal separating set for symmetric polynomials in general. We determined when is a minimal separating set for the algebra of symmetric polynomials having the least possible number of elements.
Paper Structure (6 sections, 10 theorems, 41 equations)

This paper contains 6 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Symmetric polynomials
  3. Results
  4. Auxiliaries
  5. The case of F3
  6. The general case

Key Result

lemma 1

For every $n \geq 1$ we have where

Theorems & Definitions (19)

  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proposition 1
  • proof
  • theorem 1
  • proof
  • lemma 3
  • ...and 9 more