# Rouch\'e's Theorem and the Geometry of Rational Functions

## TL;DR

This paper leverages Rouché's theorem and properties of the logarithmic derivative to derive new geometric insights into the zeros, poles, and critical points of rational functions, including improved bounds on their distances.

## Contribution

It introduces novel results on the geometry of rational functions, notably enhancing previous bounds on the proximity of zeros or poles to critical points.

## Key findings

- Improved bounds on the distance from zeros or poles to critical points
- New geometric relations involving zeros, poles, and critical points
- Application of Rouché's theorem to rational function analysis

## Abstract

In this note, we use Rouch\'e's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results regarding the geometry of the zeros, poles, and critical points of a rational function. Included is an improvement on a result by Alexander and Walsh regarding the distance from a given zero or pole of a rational function to the nearest critical point.