### A counterexample to subanalyticity of an arc-analytic function

We construct an arc-analytic function (i.e. a function analytic on each analytic arc) whose graph is not subanalytic.

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We construct an arc-analytic function (i.e. a function analytic on each analytic arc) whose graph is not subanalytic.

A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any...

A continuous linear extension operator, different from Whitney’s, for ${\mathcal{C}}^{p}$-Whitney fields (p finite) on a closed o-minimal subset of ${\mathbb{R}}^{n}$ is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.

We demonstrate that the composite function theorems of Bierstone-Milman-Pawłucki and of Glaeser carry over to any polynomially bounded, o-minimal structure which admits smooth cell decomposition. Moreover, the assumptions of the o-minimal versions can be considerably relaxed compared with the classical analytic ones.

The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory T. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory ${T}_{conv}$ (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...

A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.

On étudie certaines algèbres de fonctions analytiques réelles définies sur un ouvert $\Omega $ de ${\mathbf{R}}^{n}$. La propriété principale de ces algèbres est que tout semi-analytique de $\Omega $ défini globalement à l’aide d’un nombre fini de fonctions de $\mathcal{O}\left(\Omega \right)$, admet un nombre fini de composantes connexes. En reprenant les idées de Khovanskii (lemme de Rolle généralisé), on démontre que ces algèbres restent topologiquement noethériennes quand on leur adjoint les solutions de certaines équations différentielles du ler ordre. Par...

We construct an arc-analytic function (i.e. analytic on every real-analytic arc) in ℝ² which is analytic outside a nondiscrete subset of ℝ².

We show that a subanalytic map-germ (Rⁿ,0) → (Rⁿ,0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse.

We prove that every set definable in the structure ${\mathbb{R}}_{exp}$ can be decomposed into finitely many connected analytic manifolds each of which is also definable in this structure.

We relate the notion of arc-analyticity and the one of analyticity on restriction to polynomial arcs and we prove that in the subanalytic setting, these two notions coincide.

Given a Lipschitz stratification 𝒳 that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of 𝒳 the distance function to N is locally bi-Lipschitz trivial along N. The trivialization is obtained by integration of a Lipschitz vector field.

In 1988 it was proved by the first author that the closure of a partially semialgebraic set is partially semialgebraic. The essential tool used in that proof was the regular separation property. Here we give another proof without using this tool, based on the semianalytic L-cone theorem (Theorem 2), a semianalytic analog of the Cartan-Remmert-Stein lemma with parameters.

In this paper we construct non-trivial examples of blow-analytic isomorphisms and we obtain, via toric modifications, an inverse function theorem in this category. We also show that any analytic curve in ${\mathbb{R}}^{n},n\ge 3$, can be deformed via a rational blow- analytic isomorphism of ${\mathbb{R}}^{n}$, to a smooth analytic arc.