Extremal Cubic Inequalities of Three Variables

TL;DR

This paper characterizes all extremal nonnegative homogeneous cubic polynomials in three variables, revealing their geometric properties and establishing a connection to extremal polynomials of degree six through a specific substitution.

Contribution

It provides a complete classification of extremal elements in the cone of nonnegative cubic polynomials in three variables, including geometric and algebraic characterizations.

Findings

Extremal elements have rational curve zero loci with specific singularities.

A link between extremal cubic and degree six polynomials via a substitution.

Introduction of a new concept of infinitely near zeros for inequalities.

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$f(x,y,z) \geq 0$ 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.