On the Disk of Convergence of Algebraic Power Series

TL;DR

This paper investigates the convergence disks of algebraic power series, especially in the p-adic context, revealing differences from the complex case and providing bounds related to the series' degree.

Contribution

It clarifies the relationship between convergence disks and branch points for algebraic functions, with new results in the p-adic setting and bounds based on the series' degree.

Findings

In the complex case, the convergence disk cannot contain all branch points unless the function is rational.

In the p-adic case, the convergence disk can contain all branch points, contrary to the complex case.

Provides an upper bound for the radius of convergence based on the degree of the algebraic function.

Abstract

This paper is mainly concerned with the disk of convergence of a power series s(x) representing an algebraic function of x and specifically with the relation between this disk and the branch points of the function. We shall focus especially on the p-adic case, answering some questions of basic nature, seemingly absent from the existing literature. Our methods are simple and essentially self-contained. To illustrate the issues, recall that in the complex case it follows from standard arguments that the open disk of convergence cannot contain all the branch points of x unless the series represents a rational function. In the p-adic case, we show that the analogous assertion is not true in complete generality; but we also confirm it in a number of cases, for instance under the assumption that p is not smaller than the degree of s(x) over the field of rational functions of x. In particular this gives an upper bound for the radius of convergence which has intrinsic nature. We shall also touch several related questions.