Some Extremal Symmetric Inequalities

Abstract

Let ${\mathcal{H}}_{n,d}:=\mathbb{R}[x_1$,$\dots$, $x_n{]}_d$ 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 $\mathcal{P}(A$, $\mathcal{H}):=\{f\in \mathcal{H}$ $|$ $f(a)\ge 0$ ($\forall a\in A$)$\}$. 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 ${\mathbb{R}}_{+}:=\{x\in \mathbb{R}$, $x\ge 0\}$. Some of them provide extremal polynomials of ${\mathcal{P}}_{3,10}$.