# On the Branching Geometry of Algebraic Functions

## TL;DR

This paper presents an algorithm to analyze the branching structure of algebraic functions, combining power series expansions, analytic continuation, and numerical methods to determine their domains of analyticity.

## Contribution

It introduces a novel method using analytic continuation and the Root Test to accurately determine the regions of convergence and analyticity of algebraic function branches.

## Key findings

- Power series expansions can be computed using Newton Polygon and Laurent's Theorem.
- Analytic continuation effectively determines the domain of analyticity.
- Root Test provides numerical validation of the convergence regions.

## Abstract

This paper describes an algorithm for determining the branching geometry of algebraic functions. The graphs of these complex-valued functions have a complicated interweaving structure that can be described by analytic branches separated by singular points. Power expansions for the branches in discs centered at a point can be computed using the Newton Polygon method, and expansions around annular regions centered at the origin computed using a version of Laurent's Theorem applied to algebraic functions. However, neither of the methods enable a determination of the region of convergence of the power series. In this paper, a method using analytic continuation is used to determine the domain of analyticity for the branches, and the Root Test used to numerically check the results.