Smooth solutions of quasianalytic or ultraholomorphic equations

TL;DR

This paper proves that solutions to certain polynomial equations with quasianalytic coefficients are themselves quasianalytic functions, extending known results from real-analytic to ultraholomorphic contexts, with some limitations in non-quasianalytic cases.

Contribution

It extends the classical quasianalyticity result for solutions of polynomial equations to ultraholomorphic functions on sectors, under stability assumptions.

Findings

Solutions are in the same quasianalytic class as coefficients.

The result does not hold in general for non-quasianalytic rings.

Includes non-quasianalytic cases in ultraholomorphic setting.

Abstract

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.