# Polygons of Petrovic and Fine, algebraic ODEs, and contemporary mathematics

## TL;DR

This paper explores the historical development and significance of geometric polygon methods introduced by Petrovic and Fine for analyzing algebraic ODEs, highlighting their overlooked contributions and relevance today.

## Contribution

It uncovers and analyzes the geometric polygon methods of Petrovic and Fine, emphasizing their importance and underappreciation in modern mathematical literature.

## Key findings

- Petrovic's polygon method generalizes Newton-Puiseux techniques.
- Fine's results are better integrated into current research.
- Historical analysis reveals overlooked contributions.

## Abstract

Here, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrovic on algebraic differential equations and in particular his geometric ideas captured in his polygon method from the last years of the XIXth century, which have been left completely unnoticed by the experts. This concept, also developed in a bit a different direction and independently by Henry Fine, generalizes the famous Newton-Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrovic legacy has been practically neglected in the modern literature, while the situation is less severe in the case of results of Fine. Thus, we study the development of the ideas of Petrovic and Fine and their places in contemporary mathematics.