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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. I. Barvinok,Estimating L ∞ norms by L 2k norms for functions on orbits, Foundations of Computational Mathematics,2 (2002), 393–412.
A. Barvinok and G. Blekherman,Convex geometry of orbits, inProceedings of MSRI Workshop on Discrete Geometry, Discrete and Computational Geometry, to appear.
G. Blekherman,Convexity properties of the cone of nonnegative polynomials, Discrete and Computational Geometry32 (2004), 345–371.
L. Blum, F. Cucker, M. Shub and S. Smale,Complexity and Real Computation, Springer-Verlag, New York, 1998.
M. D. Choi, T. Y. Lam and B. Reznick,Even symmetric sextics, Mathematische Zeitschrift195 (1987), 559–580.
J. Duoandikoetxea,Reverse Hölder inequalities for spherical harmonics, Proceedings of the American Mathematical Society101 (1987), 487–491.
W. Fulton and J. Harris,Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991.
G. H. Hardy, J. E. Littlewood and G. Pólya,Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.
O. Kellogg,On bounded polynomials in several variables, Mathematische Zeitschrift27 (1928), 55–64.
M. Meyer and A. Pajor,On the Blaschke-Santal inequality, Archiv der Mathematik (Basel)55 (1990), 82–93.
J. Pach and P. Agarwal,Combinatorial Geometry, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York, 1995.
P. Parrilo,Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming96 (2003), no. 2, Series B, 293–320.
P. A. Parrilo and B. Sturmfels,Minimizing polynomials functions, inAlgorithmic and Quantitative Real Algebraic Geometry, DIMACS series in Discrete Mathematics and Theoretical Computer Science, 60, American Mathematical Society, Providence, RI, 2003, pp. 83–99.
G. Pisier,The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, 94, Cambridge University Press, Cambridge, 1989.
A. Prestel and C. Delzell,Positive Polynomials. From Hilbert’s 17th Problem to Real Algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001.
B. Reznick,Sums of even powers of real linear forms, Memoirs of the American Mathematical Society96 (1992), no. 463.
B. Reznick,Uniform denominators in Hilbert’s seventeenth problem, Mathematische Zeitschrift220 (1995), 75–97.
B. Reznick,Some concrete aspects of Hilbert’s 17th Problem, Contemporary Mathematics253 (2000), 251–272.
R. Schneider,Convex bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993.
N. Ja. Vilenkin,Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, RI, 1968.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Blekherman, G. There are significantly more nonegative polynomials than sums of squares. Isr. J. Math. 153, 355–380 (2006). https://doi.org/10.1007/BF02771790
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02771790