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/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.

中文翻译：

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一变量多值函数的解析延拓方法及其在代数方程解中的应用

本文讨论了由幂函数z = w ^{1 / ρ}生成的Puiseux级数形式在其Riemann表面的一部分上给出的一个变量的多值函数的解析连续性的几种方法，其中ρ1 /2和ρ ≠ 1.我们提出了G.Pólya定理的多页变体，描述了指数类型的所有函数的指标和共轭图之间的关系。该描述基于V.Bernstein构造的ρ阶完整函数的多页指标图≠1和普通类型。通过使用Borel方法的归纳法，可以找到“适当的” Puiseux系列的求和域（多片“ Borel多边形”）。即使在幂级数的情况下，这个结果似乎也是新的。该理论用于描述表示有理函数的逆数的Puiseux级数的解析连续性的域。结果，找到了一种求解代数方程的新方法。