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Media type:
E-Article
Title:
POSITIVE GORENSTEIN IDEALS
Contributor:
BLEKHERMAN, GRIGORIY
Published:
American Mathematical Society, 2015
Published in:
Proceedings of the American Mathematical Society, 143 (2015) 1, Seite 69-86
Language:
English
ISSN:
0002-9939;
1088-6826
Origination:
Footnote:
Description:
We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring ℝ[x] with socle in degree 2d which when viewed as a linear functional on 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.