On the convergence of formal Dulac series satisfying an algebraic ODE

TL;DR

This paper establishes a sufficient condition for the convergence of Dulac series solutions to algebraic ODEs, including those of Painlevé equations, enhancing understanding of their analytic properties.

Contribution

It introduces a new criterion that guarantees the convergence of formal Dulac series solutions to algebraic ODEs, applicable to Painlevé equations.

Findings

Proposed a sufficient convergence condition for Dulac series.

Applied the criterion to Painlevé equations, confirming convergence.

Enhanced understanding of the analytic behavior of formal solutions.

Abstract

We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and sixth Painlev\'e equations possess formal Dulac series solutions, whose convergence follows from the proposed sufficient condition.