Smooth solutions of quasianalytic or ultraholomorphic equations

Abstract

In the first part of this work, we consider a polynomial ${\varphi(x,y)=y^d+a_1(x)y^{d-1}+\cdots+a_d(x) }$ whose coefficients a _j belong to a Denjoy-Carleman quasianalytic local ring ${\mathcal{E}_1(M) }$. Assuming that ${\mathcal{E}_1(M) }$ is stable under derivation, we show that if h is a germ of C ^∞ function such that ${ \varphi(x,h(x))=0 }$, then h belongs to ${\mathcal{E}_1(M) }$. This extends a well-known fact about real-analytic functions. We also show that the result fails in general for non-quasianalytic ultradifferentiable local rings. In the second part of the paper, we study a similar problem in the framework of ultraholomorphic functions on sectors of the Riemann surface of the logarithm. We obtain a result that includes suitable non-quasianalytic situations.