Generalized Newton-Puiseux Theory and Hensel's Lemma in C[[x, y]]

Abstract

The Newton polygon and the Newton-Puiseux algorithm ([3], p. 370, [8], p. 98), and their generalizations, serve as a powerful tool for analysing the singularities of a given function. Yet experts know how difficult it is to keep track of them when one, or several, blowing-ups are applied. Thus many interesting theorems are stated under the strong, rather undesirable, assumption that the Newton faces are non-degenerate.In this paper, we introduce a method which is parallel to the classical Newton-Puiseux theory, yet avoids blowing-ups and fractional power series, except in the proofs.