Inverse Function Theorems for Arc-analytic Homeomorphisms

Details Inverse Function Theorems for Arc-analytic Homeomorphisms Inverse Function Theorems for Arc-analytic Homeomorphisms

2010-03-03

arxiv.org/abs/1003.0826v1

Mathematics

Algebraic Geometry

11 pages

Scientific paper

We call a local homeomorphism $f: (R^n,0)\to(R^n,0)$ blow-analytic if it becomes real analytic after composing with a finite number blowings-up with smooth nowhere dense centers. If the graph of $f$ is semi-algebraic then, by a theorem of Bierstone and Milman, $f$ is blow-analytic if and only if it is arc-analytic: the image by $f$ of a parametrized real analytic arc is again a real analytic arc. For a semialgebraic homeomorphism $f$ we show that if $f$ is blow-analytic and the inverse of $f$ is Lipschitz, then $f$ is Lipschitz and the inverse of $f$ is blow-analytic. The proof is by a motivic integration argument, using additive invariants on the spaces of arcs.

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