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- W2078005281 abstract "Let f (resp. φ) be a C∞ (resp. real-analytic) function germ near the origin in R n . Assume that f is divisible by φ in C∞, and that it belongs to a sufficiently regular utradifferentiable class {t! 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 d(φ) l }, where d(φ) is a suitable Lojasiewicz exponent for the regular separation between the space R 2 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." @default.
- W2078005281 created "2016-06-24" @default.
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- W2078005281 date "2002-07-01" @default.
- W2078005281 modified "2023-10-18" @default.
- W2078005281 title "A sharp division estimate for ultradifferentiable germs" @default.
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- W2078005281 doi "https://doi.org/10.2140/pjm.2002.205.237" @default.
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