Extremal Cubic Inequalities of Three Variables

Let $\mathcal{H}_{3,d}$ be the vector space of homogeneous three variable polynomials of degree $d$, and $\mathcal{P}_{3,d}^+$ be the set of all elements $f \in \mathcal{H}_{3,d}$ such that$f(x,y,z) \geq 0$ for all $x \geq 0$, $y \geq 0$, $z \geq 0$. In this article, we determine all extremal elements of $\mathcal{P}_{3,3}^+$. We prove that if $f \in \mathcal{P}_{3,3}^+$ is an irreducible extremal element, then the zero locus $V_{\mathbb{C}}(f)$ in $\mathbb{P}_{\mathbb{C}}^2$ is a rational curve whose singularity is an acnode in the interior of $\mathbb{P}_+^2$ or a cusp on an edge of $\mathbb{P}_+^2$. We also prove that if $f \in \mathcal{P}_{3,3}^+$ is an extremal element, then $f(x^2,y^2,z^2)$ is an extremal element of $\mathcal{P}_{3,6}$, where $\mathcal{P}_{3,d}$ is the set of all the elements $f \in \mathcal{H}_{3,d}$ such that $f(x,y,z) \geq 0$ for all $x$, $y$, $z \in \mathbb{R}$. A notion of infinitely near zeros of an inequality is introduced, and plays an important role.