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A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points

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Abstract

A breakthrough paper written in 1993 by Shub and Smale unveiled the relationship between stable polynomials and points which minimize the discrete logarithmic energy on the Riemann sphere (a.k.a. elliptic Fekete points). This relationship has inspired advances in the study of both concepts, many of whose main properties are not well known yet. In this paper I prove an equivalent formulation for the problem of elliptic Fekete points and some consequences, including a (nonsharp) reciprocal of Shub and Smale’s result and some novel nontrivial claims about these classical problems.

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References

  1. C. Aistleitner, J. S. Brauchart, and J. Dick, Point sets on the sphere \({\mathbb{S}}^2\) with small spherical cap discrepancy, Discrete Comput. Geom. 48 (2012), no. 4, 990–1024.

    MATH  MathSciNet  Google Scholar 

  2. D. Armentano, C. Beltrán, and M. Shub, Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2955–2965.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), no. 2, 219–245.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Beck, Sums of distances between points on a sphere-an application of the theory of irregularities of distribution to discrete geometry, Mathematika 31 (1984), no. 1, 33–41.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Beltrán, Harmonic properties of the logarithmic potential and the computability of elliptic Fekete points, Constr. Approx. 37 (2013), no. 1, 135–165.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation, Springer-Verlag, New York, 1998.

    Book  Google Scholar 

  7. J. S. Brauchart, Optimal logarithmic energy points on the unit sphere, Math. Comp. 77 (2008), no. 263, 1599–1613.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. S. Brauchart, D. P. Hardin, and E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Providence, RI, 2012, pp. 31–61.

  9. Jean-Pierre Dedieu, Estimations for the separation number of a polynomial system, J. Symbolic Comput. 24 (1997), no. 6, 683–693.

    Article  MATH  MathSciNet  Google Scholar 

  10. 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.

  11. Z. Drezner and H. W. Hamacher (eds.), Facility location, Springer-Verlag, Berlin, 2002, Applications and theory.

  12. A. Dubickas, On the maximal product of distances between points on a sphere, Liet. Mat. Rink. 36 (1996), no. 3, 303–312.

    MathSciNet  Google Scholar 

  13. M. Götz, On the distribution of weighted extremal points on a surface in \({ R}^d, d\ge 3\), Potential Anal. 13 (2000), no. 4, 345–359.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Habicht and B. L. van der Waerden, Lagerung von Punkten auf der Kugel, Math. Ann. 123 (1951), 223–234.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. B. J. Kuijlaars and E. B. Saff, Distributing many points on a Sphere, Math. Int. 19 (1997), no. 1, 5–11.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Leech, Equilibrium of sets of particles on a sphere, Math. Gaz. 41 (1957), 81–90.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Leopardi, Discrepancy, separation and Riesz energy of finite point sets on the unit sphere, Adv. Comput. Math. 39 (2013), no. 1, 27–43.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Marzo and J. Ortega-Cerdà, Equidistribution of Fekete points on the sphere, Constr. Approx. 32 (2010), no. 3, 513–521.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Letters 1 (1994), 647–662.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Reymer, Constructive theory of multivariate functions, Bibliographisches Institut, Mannheim, 1990.

    Google Scholar 

  21. M. Shub, Complexity of Bézout’s theorem. VI: Geodesics in the condition (number) metric, Found. Comput. Math. 9 (2009), no. 2, 171–178.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Shub and S. Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459–501.

    MATH  MathSciNet  Google Scholar 

  23. I. H. Sloan and R. S. Womersley, 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.

  24. I. H. Sloan and R. S. Womersley, 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.

  25. I. H. Sloan and R. S. Womersley, Extremal systems of points and numerical integration on the sphere, Adv. Comput. Math. 21 (2004), no. 1–2, 107–125.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Smale, Mathematical problems for the next century, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271–294.

    Google Scholar 

  27. J. J. Thomson, On the structure of the atom, Phil. Mag. 7 (1904), no. 6, 237–265.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. Wagner, On the product of distances to a point set on a sphere, J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 466–482.

    Article  MATH  MathSciNet  Google Scholar 

  29. G. Wagner, Erdős-Turán inequalities for distance functions on spheres, Michigan Math. J. 39 (1992), no. 1, 17–34.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. L. Whyte, Unique arrangements of points on a sphere, Amer. Math. Monthly 59 (1952), 606–611.

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I would like to thank Mike Shub for many years of friendship, collaboration, and enlightening discussions and Giuseppe Buttazzo for an inspiring remark. Thanks also to two anonymous referees for many helpful comments. Carlos Beltrán partially supported by MTM2010-16051 (Spanish Ministry of Science and Innovation MICINN).

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Correspondence to Carlos Beltrán.

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Communicated by Teresa Krick and James Renegar.

Dedicated to my dear friend Mike Shub on the occasion of his 70th birthday.

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Beltrán, C. A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points. Found Comput Math 15, 125–157 (2015). https://doi.org/10.1007/s10208-014-9213-0

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