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
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