There are significantly more nonegative polynomials than sums of squares
 Grigoriy Blekherman
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We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of even powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two, it follows that there are significantly more nonnegative polynomials than sums of squares and there are significantly more sums of squares than sums of even powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases.
 Title
 There are significantly more nonegative polynomials than sums of squares
 Journal

Israel Journal of Mathematics
Volume 153, Issue 1 , pp 355380
 Cover Date
 200612
 DOI
 10.1007/BF02771790
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 Grigoriy Blekherman ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Michigan, 481091109, Ann Arbor, MI, USA