Extreme positive ternary sextics

TL;DR

This paper characterizes specific nonnegative sextic forms in three variables that are not sums of squares, focusing on their zero sets and explicit construction methods, advancing understanding of psd forms beyond sums of squares.

Contribution

It provides a complete classification of certain non-sos sextics with nine real zeros, including explicit formulas and geometric conditions related to cubic curves and divisor classes.

Findings

Characterization of zero sets for non-sos sextics with nine points.

Explicit construction of the unique extremal non-sos sextic for given zeros.

Existence of non-sos sextics vanishing at any eight general position points.

Abstract

We study nonnegative (psd) real sextic forms $q(x_0,x_1,x_2)$ that are not sums of squares (sos). Such a form has at most ten real zeros. We give a complete and explicit characterization of all sets $S\subset\mathbb{P}^2(\mathbb{R})$ with $|S|=9$ for which there is a psd non-sos sextic vanishing in $S$. Roughly, on every plane cubic $X$ with only real nodes there is a certain natural divisor class $\tau_X$ of degree~$9$, and $S$ is the real zero set of some psd non-sos sextic if, and only if, there is a unique cubic $X$ through $S$ and $S$ represents the class $\tau_X$ on $X$. If this is the case, there is a unique extreme ray $\mathbb{R}_+q_S$ of psd non-sos sextics through $S$, and we show how to find $q_S$ explicitly. The sextic $q_S$ has a tenth real zero which for generic $S$ does not lie in $S$, but which may degenerate into a higher singularity contained in $S$. We also show that for any eight points in $\mathbb{P}^2(\mathbb{R})$ in general position there exists a psd sextic that is not a sum of squares and vanishes in the given points.