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 with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hyper-surface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of f = 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Lê et al.