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arXiv · 2105.08432

A convex form that is not a sum of squares

James Saunderson
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Abstract

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree four in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree, that are near-constant on the unit sphere, are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere.

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James Saunderson. 2021-06-11. A convex form that is not a sum of squares. https://doi.org/10.1287/moor.2022.1273

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