Solutions of quasianalytic equations

Abstract

The article develops techniques for solving equations $$G(x,y)=0$$ , where $$G(x,y)=G(x_1,\ldots ,x_n,y)$$ is a function in a given quasianalytic class (for example, a quasianalytic Denjoy–Carleman class, or the class of $${\mathcal C}^\infty $$ 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.