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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2023

Monomialization of a quasianalytic morphism

Résumé

We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes, the class of $\cC^\infty$ functions definable in a polynomially bounded $o$-minimal structure, as well as the classes of real- or complex analytic functions, and algebraic functions over any field of characteristic zero. The monomialization theorem asserts that a mapping in a quasianalytic class can be transformed to a mapping whose components are monomials with respect to suitable local coordinates, by sequences of simple modifications of the source and target -- local blowings-up and power substitutions in the real cases, in general, and local blowings-up alone in the algebraic or analytic cases. Monomialization is a version of resolution of singularities for a mapping. We show that it is not possible, in general, to monomialize by global blowings-up, even in the real-analytic case.

Dates et versions

hal-04012845 , version 1 (03-03-2023)

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Citer

André Belotto da Silva, Edward Bierstone. Monomialization of a quasianalytic morphism. Annales Scientifiques de l'École Normale Supérieure, In press. ⟨hal-04012845⟩
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