4 SHREERAM S. ABHYANKAR AND ABDALLAH ASSI
the difference sequence of m, the sequence s = (si)O:S:i:S:h+l is called the inner
product sequence of
q,
and the sequencer= (ri)o::;i:S:h+l is called the normal-
ized inner product sequence of q. Finally let Ci = mi/n for 1 :::;
i :::;
hand call
c
= (cih:S:i:S:h the normalized characteristic sequence of
f.
Given any . E
Ql,
we define the .-position of
f
to be the unique nonnegative
integer
p :::;
h such that Ci . for 1 :::;
i :::; p
and . :::;
Cj
for
p j :::;
h, and we
define the strict .-position
off
to be the unique nonnegative integer
p* :::; h
such
that ci :::; . for 1 :::;
i :::; p*
and . c1 for
p* j :::;
h. We also define the .-degree
D
(!, .) of
f
and the strict .-degree
D*
(!, .) of
f
by putting
D
(!, .) = n / dp+
1
and
D*(f,
.) = n/dp*+l· Finally we define the .-strength
S(f,
.)
off
by putting
S(f,
.) = (sp
+
(n.- mp)dp+l)fn2 if
p
=f.
0, and
S(f,
.) = . if
p
= 0.
NOTE. Note that do = 0 because the GCD of the empty set is 0. Also note that
the definition of
p
and
p*
on page 121 of
[AA]
is somewhat confusing and should
be replaced by the above definition. Finally note that, as mentioned by Zariski
on page 7 of his famous book
[Za],
in the analytic case, (mi/di+b di/di+lh:S:i:S:h
are called the characteristic pairs of
f,
and their importance was pointed out by
Smith in his 1873 paper [Sm] and by Halphen in the appendix of the 1884 French
translation [Ha] of Salmon's 1852 book [Sa] on Higher Plane Curves. For more on
characteristic pairs and the jacobian problem see
[A2]
and
[A3].
4. Contact Sets
Given any other
f'
=
f'(X, Y)
E
RQ
of Y-degree
n',
by Newton's Theorem we
can write
j'(xn',
Y)
=
IT
[Y- z.j,{X)]
l:S:j':S:n'
with
zj,(X)
E
k((X)).
The normalized contact noc(f,
f') off
and
f'
is defined
by putting
noc(f,J') = max{(1/nn'))ordx[z1(xn')- zj,(Xn)]: 1:::;
j:::;
nand 1:::;
j':::;
n'}.
Given any
F
=
F(X, Y)
E
R
of Y-degree
N,
we can write
F
=
IT
Fj
O:S:j:S:x:(F)
where
F0
=
F0 (X) E K((X))
and
F1
=
F1(X, Y) E RQ
is of Y-degree
N1
for
1 :::;
j :::; x(F),
and
x(F)
is a nonnegative integer such that:
ifF
E
k((X))
then
x(F)
= 0, whereas
ifF¢ k((X))
then
x(F)
equals the number of irreducible factors
of F in R. Upon letting
by Newton's Theorem we have
NQ
=
IT
Nj
l:S:j:S:x(F)
F(XNQ, Y)
=
F0 (XNQ)
IT
[Y- zj(X)]
l:S:j:S:N
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