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Positive Gorenstein ideals


Author: Grigoriy Blekherman
Journal: Proc. Amer. Math. Soc. 143 (2015), 69-86
MSC (2010): Primary 14N05, 14P99; Secondary 90C22, 47A57, 52A20
DOI: https://doi.org/10.1090/S0002-9939-2014-12253-2
Published electronically: August 29, 2014
MathSciNet review: 3272733
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Abstract: We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring $ \mathbb{R}[x]$ with socle in degree $ 2d$ which when viewed as a linear functional on $ \mathbb{R}[x]_{2d}$ is nonnegative on squares. Equivalently, positive Gorenstein ideals are apolar ideals of forms whose differential operator is nonnegative on squares. Positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares, and they provide a powerful framework for studying concrete aspects of sums of squares representations. We present applications of positive Gorenstein ideals in real algebraic geometry, analysis and optimization. In particular, we present a simple proof of Hilbert's nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. Drawing on our previous work (2012), our main tools are Cayley-Bacharach duality and elementary convex geometry.


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Grigoriy Blekherman
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
Email: greg@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12253-2
Received by editor(s): May 22, 2012
Received by editor(s) in revised form: May 23, 2012, and March 8, 2013
Published electronically: August 29, 2014
Communicated by: Harm Derksen
Article copyright: © Copyright 2014 American Mathematical Society