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 (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...
Consider a transitive definable action of a Lie group G on a definable manifold M. Given two (locally) definable subsets A and B of M, we prove that the dimension of the intersection σ(A) ∩ B is not greater than the expected one for a generic σ ∈ G.
We show that a subanalytic map-germ (Rⁿ,0) → (Rⁿ,0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse.
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.