Abstract:In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree $2d$. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares.
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- Grigoriy Blekherman
- Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
- MR Author ID: 668861
- Email: firstname.lastname@example.org
- Received by editor(s): December 11, 2010
- Received by editor(s) in revised form: August 12, 2011, and December 17, 2011
- Published electronically: March 15, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: J. Amer. Math. Soc. 25 (2012), 617-635
- MSC (2010): Primary 14N05, 14P99; Secondary 52A20
- DOI: https://doi.org/10.1090/S0894-0347-2012-00733-4
- MathSciNet review: 2904568