On algorithms for testing positivity of symmetric polynomial functions

TL;DR

This paper presents efficient algorithms for testing positivity of symmetric polynomial functions, including explicit discriminants for quartics, enabling polynomial-time solutions for these problems.

Contribution

It introduces polynomial-time algorithms for positivity testing of symmetric polynomials and provides explicit discriminants for symmetric quartics.

Findings

Positivity testing on symmetric polynomials is solvable in polynomial time.

Explicit discriminants for symmetric quartics are derived.

Algorithms for symmetric quartics run in linear time.

Abstract

We show that positivity on $\mathbb{R}_+^n$ and on $\mathbb{R}^n$ of real symmetric polynomials of degree at most $p$ in $n\ge2$ variables is solvable by algorithms running in $\mathrm{poly}(n)$ time. For real symmetric quartics, we find explicit discriminants and related Maple algorithms running in $\mathrm{lin}(n)$ time.