Progress report on a project whose goal is to understand the nature of singularities that have to be admitted after a blowing-up sequence that preserves the normal crossings locus of an algebraic (or complex-analytic) variety X. For example, every

surface can be transformed by blowings-up preserving normal crossings to a surface with at most additional Whitney umbrella singularities. I will discuss conjectures and recent development of techniques involving circulant matrices, Galois theory and Newton-Puiseux expansion in several variables, leading, in particular, to complete

understanding for dim X up to four. Work in collaboration with André Belotto and Ramon Ronzon Lavie.