Painleve Test and the Resolution of Singularities for Integrable Equations

TL;DR

This paper establishes a fundamental link between the Painleve test and the resolution of singularities in integrable ODEs, showing that passing the test implies the existence of a variable change that regularizes solutions.

Contribution

It proves that passing the Painleve test is equivalent to the existence of a variable transformation that converts pole singularities into regular power series solutions, ensuring convergence of principal balances.

Findings

All principal balances of an ODE system converge.

The results are consistent with Hamiltonian systems.

The Painleve test characterizes the regularity of solutions.

Abstract

We prove that under a very general setting, a system of ODE passes the Painleve test if and only if there is a good change of variable, such that the pole singularity solutions are converted to regular power series, while the converted ODE system is still kept regular. A consequence is that all principal balances of an ODE system converge. We also prove that the results are natural with respect to Hamiltonian systems.