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

aijx%V^

with

aij

- K- Let

mult(/) = min{i + j \ a^ ^ 0},

and

mult(/(0,y)) = min{j | a0j ^ 0}.

3

http://dx.doi.org/10.1090/gsm/063/02