Division and noetherianity in quasianalytic Denjoy-Carleman classes

Abstract

We show that the local rings of quasianalytic Denjoy-Carleman classes need not be Noetherian. More precisely, we construct quasianalytic Denjoy-Carleman classes $C^M$ whose local rings $C^M(\mathbb{R}^d,0)$ are not Noetherian for any $d\geq 2$. Conversely, we exhibit quasianalytic classes whose local rings in dimension two are regular and Noetherian. Our approach is based on the problem of exact division: if $f=φg$, where $f\in C^M$, $φ$ is real-analytic, and $g\in C^\infty$, under which conditions does it follow that $g\in C^M$? We introduce a quantitative scale-growth condition on the defining weight sequence $M$ and prove that exact division holds in two variables whenever this condition is sufficiently strong relative to the rate at which the nonreal branches of the complex zero set of $φ$ approach the real plane. We also construct quasianalytic weight sequences with prescribed growth thresholds and establish corresponding counterexamples to exact division. These yield nonclosed principal ideals and, consequently, non-Noetherian local rings.