Abstract : One of the possible applications of Artin approximation is to prove that the local geometry of sets defined in affine space by real or complex analytic equations is not more complicated than the local geometry of sets defined by polynomial equations. A possible approach is to prove that a complex analytic (singular) germ, for example $(X,0) \subset (\mathbf{C} ^n,0)$, is the intersection, in some affine space $\mathbf{C}^N$, of an algebraic germ $(Z,0) \subset (\mathbf{C}^N,0)$ by a complex analytic non singular subspace $(W,0)$ of dimension $n$ which is "in general position" with respect to $Z$ at the origin. Approximating $Z$ by an algebraic subspace then yields the desired result, provided the "general position" condition is sufficiently precise. I will explain how one can attack this problem using a notion of "general position with respect to a singular space" which is based on the concept of minimal Whitney stratification, which will also be explained. Nested Artin approximation is essential in this approach.

nested Artin approximation - Whitney forms - singularities - stratifications - germ of subspace