论文标题
四个变量的一些立方体和四分之一的不平等现象
Some Cubic and Quartic Inequalities of Four Variables
论文作者
论文摘要
令$ \ MATHCAL {H} \ subset \ Mathcal {h} _ {n，d}：= \ Mathbb {r} [x_1 $，$ \ ldots $，$ x_n] _d $是矢量空间，$ a $是$ a $ $ \ mathbb {p} _ {\ mathbb {r}}}^{n-1} $。我们将研究一些PSD锥$ \ MATHCAL {P} = \ MATHCAL {P}（A $，$ \ MATHCAL {H}）：= \ big \ \ {f \ in \ Mathcal {h} $ $ \ $ \ big big | $ $ f（a） Our interests are (1) to determine the extremal elements of $\mathcal{P}$, (2) to determine discriminants of $\mathcal{P}$, (3) to describe $\mathcal{P}$ as a union of basic semialgebraic subsets, and (4) to find a nice test set when $\dim \mathcal{H}$ is low.在本文中，我们介绍了（1），（2），（3）和（4），以$ \ MATHCAL {p}（\ Mathbb {r}^4 $，$ \ MATHCAL { $ \ MATHCAL {h} _ {4,4}^{s0}）$，其中$ \ MathCal {h} _ {n，d}^{s0}：= \ big \ \ \ {f \ in \ n \ natcal { $ f（1，\ ldots，1）= 0 \ big \} $。我们还提供（1） - （4）对于$ \ mathcal {p}（\ Mathbb {r} _+^4 $，$ \ nathcal {h} _ {4,3}^{c0 {c0}）$ \ Mathcal {h} _ {n，d} $ $ \ big | $ $ f $ is Cyclic and $ f（1，\ ldots，1）= 0 \ big \ \} $。
Let $\mathcal{H} \subset \mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be a vector space, and $A$ be a compact semialgebraic subset of $\mathbb{P}_{\mathbb{R}}^{n-1}$. We shall study some PSD cones $\mathcal{P} = \mathcal{P}(A$, $\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. Our interests are (1) to determine the extremal elements of $\mathcal{P}$, (2) to determine discriminants of $\mathcal{P}$, (3) to describe $\mathcal{P}$ as a union of basic semialgebraic subsets, and (4) to find a nice test set when $\dim \mathcal{H}$ is low. In this article, we present (1), (2), (3) and (4) for $\mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4}^{s0})$ and $\mathcal{P}(\mathbb{R}_+^4$, $\mathcal{H}_{4,4}^{s0})$, where $\mathcal{H}_{n,d}^{s0} := \big\{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is symmetric and $f(1,\ldots,1)=0 \big\}$. We also provide (1) -- (4) for $\mathcal{P}(\mathbb{R}_+^4$, $\mathcal{H}_{4,3}^{c0})$, where $\mathcal{H}_{n,d}^{c0} := \big\{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is cyclic and $f(1,\ldots,1)=0 \big\}$.
