Germes de d\'etermination infinie dans des classes lisses

Abstract

We give sufficient conditions of infinite determinacy (in the sense of Mather) with respect to right equivalence, in subrings of C ∞ (ℝ n ,0) defined by estimates on successive derivatives, such as rings of Gevrey germs. In this quantitative setting, a defect (hidden in the classical C ∞ case) appears between the regularity of equivalent germs and the regularity of local diffeomorphisms of (ℝn,0) giving the equivalence. Our conditions yield precise estimates of this defect, related to some suitable Łojasiewicz exponents of critical loci. We also show that the result is sharp in general.