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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Nonnegative polynomials and sums of squares
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by Grigoriy Blekherman PDF
J. Amer. Math. Soc. 25 (2012), 617-635 Request permission

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.
References
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Additional Information
  • Grigoriy Blekherman
  • Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
  • MR Author ID: 668861
  • Email: greg@math.gatech.edu
  • 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