Solution to Briot and Bouquet problem on singularities of differential equations

TL;DR

This paper solves the Briot and Bouquet problem by demonstrating the existence of non-monodromic solutions for complex differential equation singularities using simplified hedgehog dynamics.

Contribution

It provides a new, simplified approach to constructing quasi-invariant curves for indifferent irrational fixed points without complex renormalization.

Findings

Confirmed existence of non-monodromic solutions for certain singularities

Introduced a simplified method using local hedgehogs

Provided a direct construction of quasi-invariant curves

Abstract

We solve Briot and Bouquet problem (1856) on the existence of non-monodromic (multivalued) solutions for singularities of differential equations in the complex domain. The solution is an application of hedgehog dynamics for indifferent irrational fixed points. We present an important simplification by only using a local hedgehog for which we give a simpler and direct construction of quasi-invariant curves which does not rely on complex renormalization.