We give a survey of recent advances in the growth theory of entire functions associated with Pólya’s theorem on the indicator and conjugate diagrams for entire functions of exponential type. We discuss several methods of analytic continuation of a multivalued function of one variable defined on a part of its Riemann surface as a Puiseux series generated by the power function z = w1/ρ, ρ > 1/2, ρ ≠ 1. We present a multivalent variant of the Pólya theorem. The description is based on a geometric construction of Bernstein for the multivalent indicator diagram of an entire function of order ρ ≠ 1 and of normal type. We extend Borel’s method to find the region of summability for a regular Puiseux series, the multivalent Borel polygon. This result seems to be new even in the case of power series. The theory is used to describe the domains of analytic continuation for the Puiseux series representing the inverses of the rational functions. As an application, we propose a new approach to the solution of algebraic equations.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 491, 2020, pp. 94–118.
Translated by L. S. Maergoiz.
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Maergoiz, L.S. The Multivalent Indicator and Conjugate Diagrams of an Entire Function of Order ρ ≠ 1. Application to the Solution of Algebraic Equations. J Math Sci 261, 792–807 (2022). https://doi.org/10.1007/s10958-022-05789-w
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DOI: https://doi.org/10.1007/s10958-022-05789-w