We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.
A sufficient condition is proposed which ensures that a Dulac series that formally satisfies an algebraic ordinary differential equation (ODE) is convergent. Such formal solutions of algebraic ODEs are quite common: in particular, the Painlevé III, V and VI equations have formal solutions given by Dulac series; they are convergent in view of the sufficient condition presented.
Bibliography: 13 titles.
We study movable singularities of Garnier systems (and Painlevé VI equations) using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.
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