# There are significantly more nonegative polynomials than sums of squares

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## Abstract

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.

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### References

- [1]A. I. Barvinok,
*Estimating**L*^{∞}*norms by L*^{2k}*norms for functions on orbits*, Foundations of Computational Mathematics,**2**(2002), 393–412.MATHCrossRefMathSciNetGoogle Scholar - [2]A. Barvinok and G. Blekherman,
*Convex geometry of orbits*, in*Proceedings of MSRI Workshop on Discrete Geometry*, Discrete and Computational Geometry, to appear.Google Scholar - [3]G. Blekherman,
*Convexity properties of the cone of nonnegative polynomials*, Discrete and Computational Geometry**32**(2004), 345–371.MATHMathSciNetGoogle Scholar - [4]L. Blum, F. Cucker, M. Shub and S. Smale,
*Complexity and Real Computation*, Springer-Verlag, New York, 1998.Google Scholar - [5]M. D. Choi, T. Y. Lam and B. Reznick,
*Even symmetric sextics*, Mathematische Zeitschrift**195**(1987), 559–580.MATHCrossRefMathSciNetGoogle Scholar - [6]J. Duoandikoetxea,
*Reverse Hölder inequalities for spherical harmonics*, Proceedings of the American Mathematical Society**101**(1987), 487–491.MATHCrossRefMathSciNetGoogle Scholar - [7]W. Fulton and J. Harris,
*Representation Theory. A First Course*, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991.MATHGoogle Scholar - [8]G. H. Hardy, J. E. Littlewood and G. Pólya,
*Inequalities*, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.MATHGoogle Scholar - [9]O. Kellogg,
*On bounded polynomials in several variables*, Mathematische Zeitschrift**27**(1928), 55–64.CrossRefMathSciNetGoogle Scholar - [10]M. Meyer and A. Pajor,
*On the Blaschke-Santal inequality*, Archiv der Mathematik (Basel)**55**(1990), 82–93.MATHMathSciNetGoogle Scholar - [11]J. Pach and P. Agarwal,
*Combinatorial Geometry*, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York, 1995.MATHGoogle Scholar - [12]P. Parrilo,
*Semidefinite programming relaxations for semialgebraic problems*, Mathematical Programming**96**(2003), no. 2, Series B, 293–320.MATHCrossRefMathSciNetGoogle Scholar - [13]P. A. Parrilo and B. Sturmfels,
*Minimizing polynomials functions*, in*Algorithmic and Quantitative Real Algebraic Geometry*, DIMACS series in Discrete Mathematics and Theoretical Computer Science, 60, American Mathematical Society, Providence, RI, 2003, pp. 83–99.Google Scholar - [14]G. Pisier,
*The Volume of Convex Bodies and Banach Space Geometry*, Cambridge Tracts in Mathematics, 94, Cambridge University Press, Cambridge, 1989.MATHGoogle Scholar - [15]A. Prestel and C. Delzell,
*Positive Polynomials. From Hilbert’s 17th Problem to Real Algebra*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001.MATHGoogle Scholar - [16]B. Reznick,
*Sums of even powers of real linear forms*, Memoirs of the American Mathematical Society**96**(1992), no. 463.Google Scholar - [17]B. Reznick,
*Uniform denominators in Hilbert’s seventeenth problem*, Mathematische Zeitschrift**220**(1995), 75–97.MATHCrossRefMathSciNetGoogle Scholar - [18]B. Reznick,
*Some concrete aspects of Hilbert’s 17th Problem*, Contemporary Mathematics**253**(2000), 251–272.MathSciNetGoogle Scholar - [19]R. Schneider,
*Convex bodies: The Brunn-Minkowski Theory*, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993.MATHGoogle Scholar - [20]N. Ja. Vilenkin,
*Special Functions and the Theory of Group Representations*, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, RI, 1968.MATHGoogle Scholar