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 $\mathcal{P}_{3,10}$.

翻译：设 $\mathcal{H}_{n,d} := \mathbb{R}[x_1$，$\ldots$，$x_n]_d$ 为所有 $d$ 次齐次多项式的集合，并设 $\mathcal{H}_{n,d}^s := \mathcal{H}_{n,d}^{\mathfrak{S}_n}$ 为所有对称多项式的子集。对于 $\mathbb{R}^n$ 中的半代数子集 $A \subset \mathbb{R}^n$ 和向量子空间 $\mathcal{H} \subset \mathcal{H}_{n,d}$，我们通过 $\mathcal{P}(A$，$\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$ 定义PSD锥 $\mathcal{P}(A$，$\mathcal{H})$。在本文中，我们研究了 $\mathcal{P}_{3,6} := \mathcal{P}(\mathbb{R}^3$，$\mathcal{H}_{3,6})$ 和 $\mathcal{P}_{4,4} := \mathcal{P}(\mathbb{R}^4$，$\mathcal{H}_{4,4})$ 的一族极值对称多项式。我们还确定了 $\mathcal{P}_{3,5}^{s+} := \mathcal{P}(\mathbb{R}_+^3$，$\mathcal{H}_{3,5}^s)$ 的所有极值多项式，其中 $\mathbb{R}_+ := \big\{ x \in \mathbb{R}$，$x \geq 0 \big\}$。其中一些为 $\mathcal{P}_{3,10}$ 提供了极值多项式。