Analytic normal forms for convergent saddle-node vector fields

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Asymptotic expansions and summability with respect to an analytic germ

In a previous article [CMS], monomial asymptotic expansions, Gevrey asymptotic expansions, and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables - this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, for example a generalization of the Ramis-Sibuya theorem for the existence of Gevrey asymptotic expansions. Two examples of singular differential equations are presented for which the formal solutions are shown to be summable with respect to a polynomial: one ordinary and one partial differential equation

Unique solvability of coupling equations in holomorphic functions (Several aspects of microlocal analysis)

"Several aspects of microlocal analysis". October 20～24, 2014. edited by Naofumi Honda, Yasunori Okada and Susumu Yamazaki. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The theory of coupling equations was introduced by the third author [4], as a theory of a class of transformations between some nonlinear partial differential equations in complex domains. There, he constructed, to the initial value problem of a coupling equation, a formal power series solution of a special form in infinitely many variables, satisfying suitable estimates. It would be desirable, from several aspects, to study the coupling equations and their solvability as functional equations for holomorphic functions . In this report, we consider coupling equations for partial differential equations of normal form in the t variable. After preparing and recalling some notions of holomorphy on infinite dimensional spaces, we announce our recent result on the unique solvability of the initial value problem of a coupling equation, using the contraction mapping principle