Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials
HTML articles powered by AMS MathViewer
- 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.
Additional Information
- Diego Armentano
- Affiliation: Centro de Matemática, Universidad de la República, Montevideo, Uruguay
- MR Author ID: 876823
- Email: diego@cmat.edu.uy
- Carlos Beltrán
- Affiliation: Departmento de Matemáticas, Universidad de Cantabria, Santander, Spain
- MR Author ID: 764504
- ORCID: 0000-0002-0689-8232
- Email: carlos.beltran@unican.es
- Michael Shub
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada
- Address at time of publication: CONICET, Departmento de Matemáticas, Universidad de Buenos Aires, Buenos Aires, Argentina
- Email: shub.michael@gmail.com
- Received by editor(s): January 12, 2009
- Published electronically: January 11, 2011
- Additional Notes: The first author was partially supported by CSIC, Uruguay
The second author was partially suported by the research project MTM2007-62799 from the Spanish Ministry of Science MICINN
The third author was partially supported by an NSERC grant - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 2955-2965
- MSC (2010): Primary 31C20, 52A40, 60J45; Secondary 65Y20
- DOI: https://doi.org/10.1090/S0002-9947-2011-05243-8
- MathSciNet review: 2775794