Maximal contact and normal crossings in resolution of singularities

Constructive proofs of resolution of singularities over fields of characteristic zero are defined in terms of the order of ideals. However the function order has to be corrected, in order to compute centers having normal crossings with the previous exceptional divisor E. In this paper we present an algorithm of resolution of singularities (in characteristic zero), constructed using a function κ E which plays the role of the function order. The function κ E has not to be corrected since it encodes enough information of the exceptional divisor E. In fact this function allows to find regular hypersurfaces of maximal contact and having normal crossings with E.