论文标题
一些极端对称不平等
Some Extremal Symmetric Inequalities
论文作者
论文摘要
令$ \ Mathcal {h} _ {n，d}：= \ Mathbb {r} [x_1 $，$ \ ldots $，$ x_n] _d $是所有同质$ D $ d $的均值的集合，并让$ \ net $ \ mathcal {h} \ Mathcal {H} _ {n，d}^{\ Mathfrak {s} _n} $是所有对称多项式的子集。对于$ a \ subset \ mathbb {r}^n $和向量子空间$ \ Mathcal { $ \ MATHCAL {P}（A $，$ \ MATHCAL {H}）：= \ big \ {f \ in \ Mathcal {h} $ $ $ \ big | $ $ f（a）\ geq 0 $（$ \ geq 0 $（$ \ forall a \ in a a $ big \ big \} $。在本文中，我们研究了一个$ \ Mathcal {p} _ {3,6} _ {3,6}的极端对称多项式家族：= \ Mathcal {p}（\ Mathbb {r}^3 $，$ \ MATHCAL \ Mathcal {p}（\ Mathbb {r}^4 $，$ \ Mathcal {h} _ {4,4}）$。我们还确定了$ \ Mathcal {p} _ {3,5}^{s+}的所有极端多项式：= \ Mathcal {p}（\ Mathbb {r} _+^3 $，$ \ natcal {h} x \ in \ mathbb {r} $，$ x \ geq 0 \ big \} $。其中一些提供了$ \ Mathcal {p} _ {3,10} $的极端多项式。
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 $\mathcal{P}_{3,10}$.
