4

2. Non-singularity and Resolution of Singularities

Set r0 = mult(/(0, j/)) mult(/). Set

5o = min

:

: j

TQ

and an ^ 0 .

[r0-j J

£0 = oo if and only if / =

uyr°,

where u is a unit in K[[x, y]]. Suppose that

So oo. Then we can write

2+5o:/£on)

with aor0 7^ 0

a n

d the weighted leading form

Ls0(x,y) = 2^

aijx%y^

—

a0r0yr°

+ terms of lower degree in y

i+Soj=Soro

has at least two non-zero terms. We can thus choose 0 ^ c\ G K so that

Ztf0(l,ci)= ^ a y c j = 0 .

Write ^0 — ^5 where qo,Po are relatively prime positive integers. We make

a transformation

x = xf, y = xp1°(y1 + ci).

Then

/ ^ ? P ° / i ( x i , l / i ) ,

where

(2.1) /i(^i,yi) = X ]

a

^'(

C l + yi)3 +xiH(xiiVi)-

i-\-Soj=Soro

By our choice of ci, /i(0,0) = 0. Set ri = mult(/i(0, yi)). We see that

r\ TQ. We have an expansion

h = ^2aij(1)x\yv

Set

Si — min

:

: j r\ and a^-(l) 7^ 0 ,

and write 8\ = ^ with pi,gi relatively prime. We can then choose C 2 Gif

for /1 , in the same way that we chose c\ for / , and iterate this process,

obtaining a sequence of transformations

x = xf, y = x f (yi-hci),

(2.2)

x

i =

xl1*

2/i = ^(jfe + c2),

Either this sequence of transformations terminates after a finite number n

of steps with 5n = 00, or we can construct an infinite sequence of trans-

formations with Sn 00 for all n. This allows us to write y as a series in

ascending fractional powers of x.