Some Cubic and Quartic Inequalities of Four Variables

Abstract: Let $$\mathcal{H} \subset \mathcal{H}<em>{n,d} := \mathbb{R}[x_1$$,$\ldots$, $x_n]_d$ be a vector space, and $A$ be a compact semialgebraic subset of $\mathbb{P}</em>{\mathbb{R}}<sup>{n-1}$. 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 $\mathcal{P}(\mathbb{R}<sup>4$, $\mathcal{H}<em>{4,4}<sup>{s0})$ and $\mathcal{P}(\mathbb{R}</em>+<sup>4$, $\mathcal{H}<em>{4,4}<sup>{s0})$, where $\mathcal{H}</em>{n,d}^{s0} := \big{f \in \mathcal{H}<em>{n,d}$ $\big|$ $f$ is symmetric and $f(1,\ldots,1)=0 \big}$. We also provide (1) -- (4) for $\mathcal{P}(\mathbb{R}</em>+<sup>4$, $\mathcal{H}<em>{4,3}<sup>{c0})$, where $\mathcal{H}</em>{n,d}^{c0} := \big{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is cyclic and $f(1,\ldots,1)=0 \big}$.