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

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 $z=w^{1/\rho}$, where $\rho>1/2$ and $\rho\neq 1$. 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 $\rho\neq 1$ 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.