The multivalent indicator and conjugate diagrams of an entire function of order $\rho \neq 1$. Application to solution of algebraic equations

L. S. Maergoiz

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 94-118

Résumé

The paper gives a survey of recent advances in the growth theory of entire functions associated with a theorem of G. Pólya describing the relationship between 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 given on a part of its Riemann surface in the form of a Puiseux series generated by the power function $z = w^{1/\rho}$, where $\rho > 1/2,\ \rho \neq 1$. We present a multivalent variant of the mentioned theorem of G. Pólya. The description is based on a geometrical construction of V. Bernstein for the multivalent indicator diagram of an entire function of order $\rho \neq 1$ and of normal type. We extend the method of E. Borel, which allows one to find the region of summability for a “proper” Puiseux series (the multivalent Borel polygon). This result seems to be new even in the case of power series. The theory applies to description of the domains of analytic continuation for Puiseux series representing the inverse functions for rational ones. As but one consequence we elaborate a new approach to solution of algebraic equations.