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Deformations of the Zolotarev polynomials and Painlevé VI equations

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Abstract

The aim of this paper is to introduce new type of deformations of domains in the extended complex plane with a marked point and associated Green functions, the so-called iso-harmonic deformations in the first nontrivial case of doubly connected domains and to study their isomonodromic properties. We start with the Zolotarev polynomials, which are a particular case of generalized Chebyshev polynomials, namely minimal polynomials on two intervals. We introduce a deformation of elliptic curves which support Zolotarev polynomials and relate it to the Painlevé VI equations. Then, we transport these considerations into the realm of potential theory of annular domains. We deform these domains and the poles of the associated Green functions in a specific new way, by keeping invariant the corresponding harmonic measure of the boundary circles. We deduce that the critical points of the Green functions under such deformations solve a Painlevé VI equation.

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Acknowledgements

The authors would like to thank Andrey Bogatyrev for useful historical comments and the referees for the suggestions which improved the presentation. V.D. acknowledges with gratitude support from the University of Texas at Dallas, MISANU, the Serbian Ministry of Education, Science, and Technological Development, and the Science Fund of Serbia. V.S. gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada through a Discovery grant as well as from the University of Sherbrooke.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080, USA

    Vladimir Dragović

  2. Mathematical Institute SANU, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Vladimir Dragović

  3. Department of mathematics, University of Sherbrooke, 2500, boul. de l’Université, J1K 2R1, Sherbrooke, Quebec, Canada

    Vasilisa Shramchenko

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  1. Vladimir Dragović
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Correspondence to Vladimir Dragović.

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Dedicated to the memory of Boris Anatol’evich Dubrovin (1950–2019).

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Dragović, V., Shramchenko, V. Deformations of the Zolotarev polynomials and Painlevé VI equations. Lett Math Phys 111, 75 (2021). https://doi.org/10.1007/s11005-021-01415-z

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