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Analytic Continuation Methods for Multivalued Functions of One Variable and Their Application to the Solution of Algebraic Equations

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Abstract

The paper discusses several methods of analytic continuation of a multivalued function of one variable given on a part of its Riemann surface in the form of a Puiseux series generated by the power function z = w1/ρ, where ρ 1/2 and ρ ≠ 1. We present a many-sheeted variant of G.Pólya’s theorem describing the relation between the indicator and conjugate diagrams for entire functions of exponential type. The description is based on V. Bernstein’s construction for the many-sheeted indicator diagram of an entire function of order ρ ≠ 1 and normal type. The summation domain of the “proper” Puiseux series (the many-sheeted “Borel polygon”) is found with the use of a generalization of the Borel method. This result seems to be new even in the case of a power series. The theory is applied to describe the domains of analytic continuation of Puiseux series representing the inverses of rational functions. As a consequence, a new approach to the solution of algebraic equations is found.

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Acknowledgments

The author remembers Valentin Konstantinovich Ivanov with deep gratitude. It was at his seminars at Ural State University that I learned about the Pólya theorem and its impressive applications. It is my pleasant duty to express my sincere gratitude to S. R.Nasyrov and N.N.Tarkhanov for useful discussions of some questions related to this paper. The author is grateful to V.A. Stepanenko for information about the bibliography on the topic of the paper and to M.N. Zav’yalov for technical assistance in preparing the manuscript.

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Authors and Affiliations

  1. Krasnoyarsk Scientific Center of the Siberian Branch of the RAS, Krasnoyarsk, 660036, Russia

    L. S. Maergoiz

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  1. L. S. Maergoiz
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Correspondence to L. S. Maergoiz.

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In memory of Valentin Konstantinovich Ivanov on the 110th anniversary of his birth

Russian Text © The Author(s), 2019, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, Vol. 25, No. 1, pp. 120–135.

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Maergoiz, L.S. Analytic Continuation Methods for Multivalued Functions of One Variable and Their Application to the Solution of Algebraic Equations. Proc. Steklov Inst. Math. 308 (Suppl 1), 135–151 (2020). https://doi.org/10.1134/S008154382002011X

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  • DOI: https://doi.org/10.1134/S008154382002011X

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