v-Quasi-ordinary power series: Factorisation, Newton trees and resultants

This article is devoted to the study of -quasiordinary singularities. They were introduced by H. Hironaka in order to study quasiordinary singularities from the point of view of the Newton polygon. We dene Newton trees associated to power series, and the -quasiordinary singularities are the simplest case where the Newton tree is not trivial. New- ton trees generalise splice diagrams introduced by Eisenbud and Neumann for curves. We will show that for -quasiordinary singularities, there is a factorisation associated to its Newton polygon as in the case of curves. We dene the bi-coloured Newton tree of a product fg and give a sucient condition onthis coloured Newton tree so that the resultant is a monomial times a unit and we compute this resultant from the decorations of the tree. This is a generalisation of the intersection multiplicity of curves and its computation from the splice diagram.