On the Stability of Analytic Germs under Ultradifferentiable Perturbations

Let $f$ be a real-analytic function germ whose critical locus contains a given real-analytic set $X$, and let $Y$ be a germ of closed subset of $\mathbb{R}^n$ at the origin. We study the stability of $f$ under perturbations $u$ that are flat on $Y$ and that belong to a given Denjoy-Carleman non-quasianalytic class. We obtain a condition ensuring that $f+u=f\circ\Phi$ where $\Phi$ is a germ of diffeomorphism whose components belong to a (generally larger) Denjoy-Carleman class. Roughly speaking, this condition involves a \L ojasiewicz-type separation property between $Y$ and the complex zeros of a certain ideal associated with $f$ and $X$. The relationship between the Denjoy-Carleman classes of $u$ and $\Phi$ is controlled precisely by the inequality. This result extends, and simplifies, former work of the author on germs with isolated critical points.