2015/08/16 by Aaron Kunert, Kunert, Aaron, Claus Scheiderer +1
Mathematics · #14C22 #14H45 (secondary) #14P05 (primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C22 #msc:14H45 #msc:14P05
paper · pdf · doi:10.48550/arxiv.1508.03816
v2: Error in bibliography corrected
arxiv created 2015/08/18 · arxiv updated 2015/08/19
We study nonnegative (psd) real sextic forms q(x0,x1,x2) 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⊂ℙ2(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 τX 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 τX on X. If this is the case, there is a unique extreme ray ℝ+qS of psd non-sos sextics through S, and we show how to find qS explicitly. The sextic qS 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 ℙ2(ℝ) in general position there exists a psd sextic that is not a sum of squares and vanishes in the given points.