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Constructive Approximation

, Volume 46, Issue 1, pp 37–45 | Cite as

How Many Zolotarëv Fractions are There?

  • A. B. Bogatyrëv
Article

Abstract

Known properties of the Chebyshev polynomials are the following: they have simple critical points and only two (finite) critical values. Those properties uniquely determine the polynomials modulo affine transformations of dependent and independent variables. Zolotarëv fractions have similar properties: simple critical points with only four distinct critical values. These properties determine many classes of rational functions modulo projective transformations of the dependent and independent variables. In this paper, we explicity describe these classes.

Keywords

Zolotarev fraction Elliptic curves Lattices 

Mathematics Subject Classification

30Cxx 30E10 14H52 33E05 

Notes

Acknowledgements

The author thanks Alex Eremenko, George Shabat, and Nickolai Adrianov for discussing various aspects of the paper and the anonymous referee for valuable remarks that improved the exposition.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute for Numerical MathematicsRussian Academy of SciencesMoscowRussia

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