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Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others wegeneralize the inverse and implicit function theorem (in differential, analyticand algebraic setting) to sets of functions of larger multiplicities (orideals). This allows one to describe singularities given by a finite set ofgenerators or by ideals in a simpler form. In the special Cohen-Macaulay casewe obtain a singular analog of the inverse function theorem. The singularimplicit function theorem is closely related to a (proven here) extendedversion of the Weierstrass-Hironaka-Malgrange division and preparationtheorems. The primary motivation for this paper comes from thedesingularization problem. As an illustration of the techniques used, we givesome applications of our theorems to desingularization extending some resultson Hironaka normal flatness, the Samuel stratification and the Hilbert-Samuelfunction. The notion of the standard basis along Samuel stratum introduced inthe paper (inspired by the Bierstone-Milman and Hironaka constructions) allowsus to describe singularities along the Samuel stratum in a relatively simpleway. It leads to a canonical reduction of the strong Hironaka desingularizationwith normally flat centers to a so called resolution of marked ideals.Moreover, in characteristic zero, the standard basis along Samuel stratumgenerates a unique canonical Rees algebra along Samuel startum giving astraightforward proof of the strong desingularization in algebraic and analyticcases.

Singular implicit and inverse function theorems. Strong resolution with normally flat centers