Extremal Cubic Inequalities of Three Variables

Abstract

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$f(x,y,z)\ge 0$ for all $x\ge 0$, $y\ge 0$, $z\ge 0$. 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 $f(x,y,z)\ge 0$ for all $x$, $y$, $z\in \mathbb{R}$. A notion of infinitely near zeros of an inequality is introduced, and plays an important role.