Analytic and Nash equivalence relations of Nash maps

Let $M$ and $N$ be Nash manifolds, and $f$ and $g$ Nash maps from $M$ to $N$. If $M$ and $N$ are compact and if $f$ and $g$ are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, $C^infty$ R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash map germs and analytic map germs. Indeed, there are two analytic map germs from $(R^2,0)$ to $(R^4,0)$ which are $C^infty$ R-L equivalent but not analytically R-L equivalent

Analytic equivalence of normal crossing functions on a real analytic manifold

By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a suitable modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions $C^{\infty}$ right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of equivalence classes is zero or countable

O-minimal Hauptvermutung for polyhedra II

Hilbert initiated the standpoint in foundations of mathematics. From this standpoint, we allow only a finite number of repetitions of elementary operations when we construct objects and morphisms. When we start from a subset of a Euclidean space. Then we assume that any element of the line has only a finite number of connected components. We call the set tame if the assumption is satisfied, and define a tame morphism in the same way. In this paper we will show that a tame topological manifold is carried by a tame homeomorphism to the interior of a compact piecewise linear manifolds possibly with boundary and such a piecewise linear manifold possibly with boundary is unique up to piecewise linear homeomorphisms in the sense that if two manifolds are such PL manifolds possibly with boundary then they are the same as piecewise linear manifolds. We modify this to Theorem 2 so that argument of model theory works, and we prove it. We also consider the differentiable case.Comment: 45 page

Artin approximation compatible with a change of variables

We propose a version of the classical Artin approximation which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a Nash equation by a Nash solution in a compatible way with a given Nash change of variables. This results is closely related to the so-called nested Artin approximation and becomes false in the analytic setting. We provide local and global version of this approximation in real and complex geometry together with an application to the Right-Left equivalence of Nash maps