Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials
Authors:
Diego Armentano, Carlos Beltrán and Michael Shub
Journal:
Trans. Amer. Math. Soc. 363 (2011), 29552965
MSC (2010):
Primary 31C20, 52A40, 60J45; Secondary 65Y20
Published electronically:
January 11, 2011
MathSciNet review:
2775794
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove that points in the sphere associated with roots of random polynomials via the stereographic projection are surprisingly wellsuited 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.
 1.
C. Beltrán and L.M. Pardo, Fast linear homotopy to find approximate zeros of polynomial systems, To appear (2008).
 2.
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
(2009m:90147), 10.1090/S0894034708006309
 3.
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 2sphere, J. Comput. Phys. 228
(2009), no. 9, 3288–3306. MR 2513833
(2010d:65399), 10.1016/j.jcp.2009.01.021
 4.
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.
 5.
Lenore
Blum, Felipe
Cucker, Michael
Shub, and Steve
Smale, Complexity and real computation, SpringerVerlag, New
York, 1998. With a foreword by Richard M. Karp. MR 1479636
(99a:68070)
 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 (2003h:41020)
 7.
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
(98e:11092), 10.1090/S0002994798021199
 8.
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
(96e:78011), 10.4310/MRL.1994.v1.n6.a3
 9.
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
(2010f:65103), 10.1007/s1020800790176
 10.
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
(93k:65045), 10.1090/S08940347199311759804
 11.
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
(94m:68086)
 12.
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
(94g:65152), 10.1006/jcom.1993.1002
 13.
Steve
Smale, Mathematical problems for the next century,
Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI,
2000, pp. 271–294. MR 1754783
(2001i:00003)
 14.
L.
L. Whyte, Unique arrangements of points on a sphere, Amer.
Math. Monthly 59 (1952), 606–611. MR 0050303
(14,310c)
 15.
Qi
Zhong, Energies of zeros of random sections on Riemann
surfaces, Indiana Univ. Math. J. 57 (2008),
no. 4, 1753–1780. MR 2440880
(2009k:58051), 10.1512/iumj.2008.57.3329
 16.
Y. Zhou, Arrangements of points on the sphere, Ph.D. Thesis. Math. Department, University of South Florida, 1995.
 1.
 C. Beltrán and L.M. Pardo, Fast linear homotopy to find approximate zeros of polynomial systems, To appear (2008).
 2.
 , Smale's 17th problem: Average polynomial time to compute affine and projective solutions, J. Amer. Math. Soc. 22 (2009), 363385. MR 2476778 (2009m:90147)
 3.
 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 sphere, J. Comput. Phys. 228 (2009), no. 9, 32883306. MR 2513833
 4.
 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.
 5.
 L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and real computation, SpringerVerlag, New York, 1998. MR 1479636 (99a:68070)
 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. 137144. MR 1924855 (2003h:41020)
 7.
 A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350 (1998), 523538. MR 1458327 (98e:11092)
 8.
 E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Letters 1 (1994), 647662. MR 1306011 (96e:78011)
 9.
 M. Shub, Complexity of Bézout's theorem. VI: Geodesics in the condition (number) metric, Found. Comput. Math. 9 (2009), no. 2, 171178. MR 2496558
 10.
 M. Shub and S. Smale, Complexity of Bézout's theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459501. MR 1175980 (93k:65045)
 11.
 , Complexity of Bézout's theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992), Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267285. MR 1230872 (94m:68086)
 12.
 , Complexity of Bézout's theorem. III. Condition number and packing, J. Complexity 9 (1993), no. 1, 414, Festschrift for Joseph F. Traub, Part I. MR 1213484 (94g:65152)
 13.
 S. Smale, Mathematical problems for the next century, Mathematics: Frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271294. MR 1754783 (2001i:00003)
 14.
 L. L. Whyte, Unique arrangements of points on a sphere, Amer. Math. Monthly 59 (1952), 606611. MR 0050303 (14:310c)
 15.
 Q. Zhong, Energies of zeros of random sections on Riemann surfaces, Indiana Univ. Math. J. 57 (2008), no. 4, 17531780. MR 2440880 (2009k:58051)
 16.
 Y. Zhou, Arrangements of points on the sphere, Ph.D. Thesis. Math. Department, University of South Florida, 1995.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
31C20,
52A40,
60J45,
65Y20
Retrieve articles in all journals
with MSC (2010):
31C20,
52A40,
60J45,
65Y20
Additional Information
Diego Armentano
Affiliation:
Centro de Matemática, Universidad de la República, Montevideo, Uruguay
Email:
diego@cmat.edu.uy
Carlos Beltrán
Affiliation:
Departmento de Matemáticas, Universidad de Cantabria, Santander, Spain
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
DOI:
http://dx.doi.org/10.1090/S000299472011052438
Keywords:
Logarithmic energy,
elliptic Fekete points,
random polynomials
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 MTM200762799 from the Spanish Ministry of Science MICINN
The third author was partially supported by an NSERC grant
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
