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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Extreme positive ternary sextics


Authors: Aaron Kunert and Claus Scheiderer
Journal: Trans. Amer. Math. Soc. 370 (2018), 3997-4013
MSC (2010): Primary 14P05; Secondary 14C22, 14H45
DOI: https://doi.org/10.1090/tran/7076
Published electronically: December 14, 2017
MathSciNet review: 3811517
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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.


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Additional Information

Aaron Kunert
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, Germany

Claus Scheiderer
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, Germany
MR Author ID: 212893

Received by editor(s): August 27, 2015
Received by editor(s) in revised form: September 8, 2016
Published electronically: December 14, 2017
Article copyright: © Copyright 2017 American Mathematical Society