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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
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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\subset \mathbb{P}^2(\mathbb{R})$ with $ \vert S\vert=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

DOI: https://doi.org/10.1090/tran/7076
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

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