2.1. Newton's method for determining the branches of a plane curve 5

As our first approximation, we can use our first transformation to solve

for y in terms of x and y\:

y

=

Clx5°

+

yix6°.

Now the second transformation gives us

y

=

Clx6°

+

c2xSo+^

+

y2x6o+^.

We can iterate this procedure to get the formal fractional series

(2.3) y =

cix6°

+

c2xSoJr^

+ c

3

/

0 +

^

+

^ T + • • • .

Theorem 2.1. There exists an io such that Si £ N for i io.

Proof, ri = mult(/i(0,?/i)) are monotonically decreasing, and positive for

all i, so it suffices to show that n = n+i implies (SjGN. Without loss of

generality, we may assume that i = 0 and r*o = r\. /i(#i, yi) is given by the

expression (2.1). Set

g(t) = hfat) = Y, M*+*)'"•

g(t) has degree r$. Since r\ = ro, we also have mult (#(£)) = ro- Thus

g(t) =

a0rotr°,

and

] P a ^ =a

0

ro(^-ci) r o .

In particular, since if has characteristic 0, the binomial theorem shows that

(2.4) ev

0

_i ^ 0,

where i is a natural number with i + So{ro — 1) = Soro- Thus 5 o G N. •

We can thus find a natural number ra, which we can take to be the

smallest possible, and a series

p(t) = X *

such that (2.3) becomes

(2.5) y=p(X™).

For n £ N, set

Using induction, we can show that

Pn{t) =

J2bif'

i = l

tha t

mult(/(t

m

,p

n

(t))-oo