Some Extremal Symmetric Inequalities

Let $\mathcal{H}_{n,d}^s := \mathbb{R}[x_1$,$\ldots$, $x_n]_d^{\mathfrak{S}_n}$ be the set of symmetric polynomials of degree $d$, and $A$ be a compact semialgebraic subset of $\mathbb{P}_{\mathbb{R}}^{n-1}$. We shall study some PSD cones $\mathbb{P}(A$, $\mathcal{H}_{n,d}^s) := \big\{f \in \mathcal{H}_{n,d}^s$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. In this article, we study extremal elements of $\mathcal{P}(\mathbb{R}_+^3$, $\mathcal{H}_{3,5}^{s+})$, $\mathcal{P}(\mathbb{R}_+^3$, $\mathcal{H}_{3,6}^{s0})$, and $\mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4}^s)$ where $\mathcal{P}(\mathbb{R}_+^3$, $\mathcal{H}_{3,6}^{s0}) := \big\{f \in \mathcal{P}(\mathbb{R}_+^3$, $\mathcal{H}_{3,6}^s)$ $\big|$ $f(1,\ldots,1)=0 \big\}$.