Flatness in analytic mappings I. On an announcement of Hironaka

We present a stratification by “normal flatness” associated to an analytic mapping, analogous to Hironaka's classical result for analytic spaces. Our construction is based on a generic normal flatness theorem for mappings, proved using techniques concerning the variation of modules of meromorphically parametrized formal power series 1]. The existence of such a stratification was announced by Hironaka 13], but the other claims made in 13] are false. Counterexamples are also presented here.