Some Extremal Symmetric Inequalities

TL;DR

This paper investigates extremal symmetric polynomials within certain positive semidefinite cones, characterizing their structure and identifying all extremal cases for specific polynomial spaces and domains.

Contribution

It characterizes extremal symmetric polynomials in semialgebraic cones, providing explicit descriptions for several polynomial spaces and domains, including extremal polynomials in positive orthants.

Findings

Identified extremal symmetric polynomials in ,6 and ,4 polynomial spaces.

Determined all extremal polynomials in ,5^{s+} domain.

Some extremal polynomials extend to ,10 cases.

Abstract

Let $\mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be the set of all the homogeneous polynomials of degree $d$, and let $\mathcal{H}_{n,d}^s := \mathcal{H}_{n,d}^{\mathfrak{S}_n}$ be the subset of all the symmetric polynomials. For a semialgebraic subset of $A \subset \mathbb{R}^n$ and a vector subspace $\mathcal{H} \subset \mathcal{H}_{n,d}$, we define a PSD cone $\mathcal{P}(A$, $\mathcal{H})$ by $\mathcal{P}(A$, $\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. In this article, we study a family of extremal symmetric polynomials of $\mathcal{P}_{3,6} := \mathcal{P}(\mathbb{R}^3$, $\mathcal{H}_{3,6})$ and that of $\mathcal{P}_{4,4} := \mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4})$. We also determine all the extremal polynomials of $\mathcal{P}_{3,5}^{s+} := \mathcal{P}(\mathbb{R}_+^3$, $\mathcal{H}_{3,5}^s)$ where $\mathbb{R}_+ := \big\{ x \in \mathbb{R}$, $x \geq 0 \big\}$. Some of them provide extremal polynomials of $\mathcal{P}_{3,10}$.