Smooth solutions of quasianalytic or ultraholomorphic equations

In the first part of this work, we consider a polynomial $\varphi (x,y)=y^d+a_1(x)y^{d-1}+\mathrm{.}\mathrm{.}\mathrm{.}+a_d(x)$ 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 $C^{\mathrm{\infty }}$ 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.