Asymptotic property of divergent formal solutions in linearization of singular vector fields (Q653286)

The author studies some asymptotic properties of divergent formal solutions that appear in the linearization problem of a singular vector field without assuming a Diophantine condition or the existence of additional first integrals. In the linearization problem of a singular vector field a linearizing transformation should satisfy a certain semilinear Fuchsian system of differential equations of several variables. Although the system has a formal power series solution under an appropriate nonresonance condition, such a solution does not converge in general unless we assume a Diophantine condition or the existence of additional first integrals. In this paper, introducing a singular perturbative parameter \(\epsilon\) to the system of equations of two independent variables (in such a way that \(\epsilon=1\) corresponds to the original system), the author constructs a singular perturbative solution (i.e., formal power series solution in \(\epsilon\)) and studies its asymptotic properties through the Borel resummation technique without assuming a Diophantine condition or the existence of additional first integrals. To be more specific, it is shown that, if the nonlinear part of the system satisfies certain support conditions, the resummed singular perturbative solution thus constructed can be analytically continued with respect to \(\epsilon\) to a sector with vertex at \(\epsilon=1\) and gives there an asymptotic expansion of a certain analytic solution of the system in a multisector of the space variables.