Existence of non-algebraic singularities of differential equation

TL;DR

This paper proves that there are infinitely many saddle-node singularities in the complex plane that cannot be described by algebraic differential equations, highlighting limitations in algebraic approaches to singularities.

Contribution

It demonstrates the existence of countably many non-algebraic saddle-node singularities, expanding understanding of singularities beyond algebraizable cases.

Findings

Existence of countably many non-algebraic saddle-node singularities

Not all singularities can be described by algebraic differential equations

Limits of algebraic methods in foliation singularity classification

Abstract

An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many saddle-node singularities of the complex plane that are not algebraizable.