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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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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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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DOI: https://doi.org/10.1007/s10208-014-9213-0
Keywords
- Elliptic Fekete points
- Stable polynomials
- Logarithmic energy
- Smale’s 7th problem
- Facility location
- Cap discrepancy