Decomposition in bunches of the critical locus of a quasi-ordinary map

Abstract

A polar hypersurface P of a complex analytic hypersurface germ, f=0, can be investigated by analyzing the invariance of certain Newton polyhedra associated to the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated to f. We develop this general principle of Teissier (see Varietes polaires. I. Invariants polaires des singularites d'hypersurfaces, Invent. Math. 40 (1977), 3, 267-292) when f=0 is a quasi-ordinary hypersurface germ and P is the polar hypersurface associated to any quasi-ordinary projection of f=0. We build a decomposition of P in bunches of branches which characterizes the embedded topological type of the irreducible components of f=0. This decomposition is characterized also by some properties of the strict transform of P by the toric embedded resolution of f=0 given by the second author in a paper which will appear in Annal. Inst. Fourier (Grenoble). In the plane curve case this result provides a simple algebraic proof of the main theorem of Le, Michel and Weber in "Sur le comportement des polaires associees aux germes de courbes planes", Compositio Math, 72, (1989), 1, 87-113.