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
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