Characteristic Pairs along the Resolution Sequence

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Suppose that f is irreducible in a power series ring in two variables over an algebraically closed field k of characteristic 0. The characteristic pairs of f can be defined from a fractional power series expansion of a solution of f . The singularity of f can be resolved by a finite number of blow ups of points. This subject, which can be traced back to Newton, has been studied extensively. A few references are Abhyankar [1], Brieskorn and Knorrer [2], Campillo [3], Enriques and Chisini [4] and Zariski [7]. In Sections 1 and 2 we give an exposition of the basic results in the theory of Puiseux series. In Section 3 we give a formula for the characteristic pairs of the transform of f along the sequence of blow ups of points resolving the singularity. As a corollary, we obtain the classical theorem of Enriques and Chisini relating the multiplicity sequence of a resolution and the characteristic pairs of f , and we recover the classical result that the characteristic pairs are an invariant of f . We use an inversion formula of Abhyankar to obtain the results of this paper. 1. The Puiseux series. Let R be a power series ring in two variables over an algebraically closed field k. Then we have the following well-known theorem (see [2, pp. 405 406], [7, p. 7]). Theorem 1.1. Suppose that f ∈ R is irreducible and (x, y) are regular parameters for R such that the multiplicity ν(f) = ν(f(0, y)). Then a fractional power series exists (called a Puiseux series) of y in terms of x. The expansion has the form