Some Cubic and Quartic Inequalities of Four Variables

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 $f(1,\ldots,1)=0 \big\}$.