Submission date: 22 July 2019

Abstract:

We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes (of interest in real analysis), the class of infinitely differentiable functions which are definable in a given polynomially bounded o-minimal structure (in model theory), 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 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. It is not possible, in general, to monomialize by global blowings-up, even in the real analytic case.

Mathematics Subject Classification: rimary 03C64, 14E05, 26E10, 32S45, Secondary 03C10, 14E15, 30D60, 32B20

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Full text arXiv 1907.09502: pdf, ps.