Extreme positive ternary sextics

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 $S\subseteq \mathbb {P}^2(\mathbb {R})$ 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.