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.
A decomposition of a set definable in an o-minimal structure into perfectly situated sets
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 linear extension operator for Whitney fields on closed o-minimal sets
A continuous linear extension operator, different from Whitney’s, for -Whitney fields (p finite) on a closed o-minimal subset of is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.
A local Bernstein inequality on real algebraic varieties.
A note on Bierstone-Milman-Pawłucki's paper "Composite differentiable functions"
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.
A proof of the valuation property and preparation theorem
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 (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...
A remark on the Lion-Rolin Preparation Theorem for LA-functions
A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.
Algèbres analytiques topologiquement noéthériennes. Théorie de Khovanskii
On étudie certaines algèbres de fonctions analytiques réelles définies sur un ouvert de . La propriété principale de ces algèbres est que tout semi-analytique de défini globalement à l’aide d’un nombre fini de fonctions de , 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...
An arc-analytic function with nondiscrete singular set
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.
An inverse mapping theorem for arc-analytic homeomorphisms
We show that a subanalytic map-germ (Rⁿ,0) → (Rⁿ,0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse.
Analytic cell decomposition of sets definable in the structure
We prove that every set definable in the structure can be decomposed into finitely many connected analytic manifolds each of which is also definable in this structure.
Arc-analyticity and polynomial arcs
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.
Artin-Large property for analytic manifolds of dimension two.