Analytic reparametrization of semi-algebraic sets

In many problems in analysis, dynamics, and in their applications, it is important to subdivide objects under consideration into simple pieces, keeping control of high-order derivatives. It is known that semi-algebraic sets and mappings allow for such a controlled subdivision: this is the “Ck reparametrization theorem” which is a high-order quantitative version of the well-known results on the existence of a triangulation of semi-algebraic sets. In a Ck-version we just require in addition that each simplex be represented as an image, under the “reparametrization mapping” ψ, of the standard simplex, with all the derivatives of ψ up to order k uniformly bounded. The main result of this paper is, that if we reparametrize all the set A but its small part of a size δ, we can do much more: not only to “kill” the derivatives, but also to bound uniformly the analytic complexity of the pieces, while their number remains of order .