Abstract

Let f (resp. ϕ) be a C^∞ (resp. real-analytic) function germ near the origin in Rⁿ. Assume that f is divisible by ϕ in C^∞, and that it belongs to a suficiently regular utradifferentiable class {ℓ!M_ℓ} of Carleman type (for example, one of the Gevrey rings familiar in the theory of differential equations). What can then be said about the regularity of the quotient f ∕ ϕ? In this paper, we obtain first a complete solution of this problem in the case n = 2. Namely, it is shown that f ∕ ϕ belongs to the Carleman class {ℓ!M_ℓ^d(ϕ)}, where d(ϕ) is a suitable Łojasiewicz exponent for the regular separation between the space R² and certain components of the complex zero set Z_ϕ of ϕ. This number can be explicitely computed by means of Puiseux expansions. We prove moreover that the division result is sharp for any ϕ and M. Finally, we apply it to get a characterization of closed principal ideals generated by real-analytic functions in Carleman classes of two variables, improving a result which was known previously only in the case of generators with isolated real zeros.

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