Solutions of quasianalytic equations

TL;DR

This paper develops methods for solving quasianalytic equations, showing that formal solutions correspond to actual solutions with controlled regularity loss, and discusses related fundamental properties of quasianalytic functions.

Contribution

It establishes a link between formal power series solutions and actual quasianalytic solutions, extending classical results to quasianalytic classes with controlled regularity loss.

Findings

Formal solutions correspond to actual solutions with regularity loss

Framework applies to division, factorization, Weierstrass preparation in quasianalytic classes

Provides techniques for solving quasianalytic equations

Abstract

The article develops techniques for solving equations G(x,y)=0, where G(x,y)=G(x_1,...,x_n,y) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class, or the class of infinitely differentiable functions definable in a polynomially-bounded o-minimal structure). We show that, if G(x,y)=0 has a formal power series solution y=H(x) at some point a, then H is the Taylor expansion at a of a quasianalytic solution y=h(x), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.