Chapter 2 Non-singularity and Resolution of Singularities 2.1. Newton's method for determining the branches of a plane curve Newton's algorithm for solving f(x,y) = 0 by a fractional power series y = j/(xm) can be thought of as a generalization of the implicit function theorem to general analytic functions. We begin with this algorithm because of its simplicity and elegance, and because this method contains some of the most important ideas in resolution. We will see (in Section 2.5) that the algorithm immediately gives a local solution to resolution of analytic plane curve singularities, and that it can be interpreted to give a global solution to resolution of plane curve singularities (in Section 3.5). All of the proofs of resolution in this book can be viewed as generalizations of Newton's algorithm, with the exception of the proof that curve singularities can be resolved by normalization (Theorems 2.14 and 4.3). Suppose K is an algebraically closed field of characteristic 0, K[[x, y}] is a ring of power series in two variables and / G if[[x,y]] is a non-unit, such that x | / . Write / = Ylij a ijx%V^ with a ij - K- Let mult(/) = min{i + j \ a^ ^ 0}, and mult(/(0,y)) = min{j | a0j ^ 0}. 3
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