A note on the Weierstrass preparation theorem in quasianalytic local rings (Q2925380)

The main result proved in the paper is that if a quasianalytic system \(\mathcal{C} = \{ \mathcal{C}_n : n \in \mathbb{N}\}\) (i.e., a system of local rings of germs of smooth quasianalytic functions containing all germs of analytic functions and closed under composition, implicit equation, and monomial division) satisfies a certain Weierstrass preparation property, then it is contained in the analytic system. This extends a bit a result of \textit{A. Elkhadiri} and \textit{H. Sfouli} [Stud. Math. 185, No. 1, 83--86 (2008; Zbl 1144.26032)]. A similar kind of result was announced by \textit{F. Acquistapace, F. Broglia, M. Bronshtein, A. Nicoara} and \textit{N. Zobin} [Adv. Math. 258, 397--413 (2014; Zbl 1292.26069)], although the approach in this latter paper is different. Quasianalytic means that for each \(n \in \mathbb{N}\) the map which associates to \(f\) its Taylor expansion at the origin is injective on \(\mathcal{C}_n\). The study of quasianalytic functions and classes dates back to work of E. Borel, J. Hadamard, A. Denjoy and T. Carleman in the early part of the 20th century.