Maillet--Malgrange type theorem for a formal Dulac series solution of an analytic ODE

TL;DR

This paper proves a Maillet-Malgrange type theorem for Dulac series solutions of analytic ODEs, enabling estimation of coefficient growth and providing conditions for convergence, thus advancing understanding of formal solutions in complex dynamics.

Contribution

It extends the Maillet-Malgrange theorem to Dulac series with complex exponents, offering new tools to analyze their growth and convergence properties.

Findings

Establishes bounds on the growth of Dulac series coefficients.

Provides criteria for the convergence of Dulac series solutions.

Determines the Gevrey order of formal Dulac series.

Abstract

A Maillet-Malgrange type theorem is proved for a Dulac series (in the general case, with complex exponents), which formally satisfies an analytical ordinary differential equation (ODE). This theorem allows to estimate the growth of the norms of the coefficients of such a series, that is, to determine its Gevrey order, and in the special case it provides a sufficient condition for the convergence of the series.