Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials

Armentano, Diego; Beltrán, Carlos; Shub, Michael

doi:10.1090/S0002-9947-2011-05243-8

Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials
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by Diego Armentano, Carlos Beltrán and Michael Shub PDF

Trans. Amer. Math. Soc. 363 (2011), 2955-2965 Request permission

Abstract:

We prove that points in the sphere associated with roots of random polynomials via the stereographic projection are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere. That is, roots of random polynomials provide a fairly good approximation to elliptic Fekete points.

References

C. Beltrán and L.M. Pardo, Fast linear homotopy to find approximate zeros of polynomial systems, To appear (2008).
Carlos Beltrán and Luis Miguel Pardo, Smale’s 17th problem: average polynomial time to compute affine and projective solutions, J. Amer. Math. Soc. 22 (2009), no. 2, 363–385. MR 2476778, DOI 10.1090/S0894-0347-08-00630-9
E. Bendito, A. Carmona, A. M. Encinas, J. M. Gesto, A. Gómez, C. Mouriño, and M. T. Sánchez, Computational cost of the Fekete problem. I. The forces method on the 2-sphere, J. Comput. Phys. 228 (2009), no. 9, 3288–3306. MR 2513833, DOI 10.1016/j.jcp.2009.01.021
E. Bendito, A. Carmona, A.M. Encinas, and J.M. Gesto, Computational cost of the Fekete Problem II: on Smale’s 7th Problem, To appear.
Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale, Complexity and real computation, Springer-Verlag, New York, 1998. With a foreword by Richard M. Karp. MR 1479636, DOI 10.1007/978-1-4612-0701-6
P. D. Dragnev, On the separation of logarithmic points on the sphere, Approximation theory, X (St. Louis, MO, 2001) Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 137–144. MR 1924855
A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350 (1998), no. 2, 523–538. MR 1458327, DOI 10.1090/S0002-9947-98-02119-9
E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Lett. 1 (1994), no. 6, 647–662. MR 1306011, DOI 10.4310/MRL.1994.v1.n6.a3
Michael Shub, Complexity of Bezout’s theorem. VI. Geodesics in the condition (number) metric, Found. Comput. Math. 9 (2009), no. 2, 171–178. MR 2496558, DOI 10.1007/s10208-007-9017-6
Michael Shub and Steve Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459–501. MR 1175980, DOI 10.1090/S0894-0347-1993-1175980-4
M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267–285. MR 1230872, DOI 10.1007/978-1-4612-2752-6_{1}9
Michael Shub and Steve Smale, Complexity of Bezout’s theorem. III. Condition number and packing, J. Complexity 9 (1993), no. 1, 4–14. Festschrift for Joseph F. Traub, Part I. MR 1213484, DOI 10.1006/jcom.1993.1002
Steve Smale, Mathematical problems for the next century, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271–294. MR 1754783
L. L. Whyte, Unique arrangements of points on a sphere, Amer. Math. Monthly 59 (1952), 606–611. MR 50303, DOI 10.2307/2306764
Qi Zhong, Energies of zeros of random sections on Riemann surfaces, Indiana Univ. Math. J. 57 (2008), no. 4, 1753–1780. MR 2440880, DOI 10.1512/iumj.2008.57.3329
Y. Zhou, Arrangements of points on the sphere, Ph.D. Thesis. Math. Department, University of South Florida, 1995.