Smooth solutions of quasianalytic or ultraholomorphic equations

In the first part of this work, we consider a polynomial $\phi(x,y)=y^d+a_1(x)y^{d-1}+...+a_d(x)$ 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 $C^\infty$ function such that $\phi(x,h(x))=0$, 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.