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(1) f (X(t),Y(t)) = 0. (Here, f is thought of as a polynomial over the field k((t))).
(2) Not both X(t) and Y(t) belong to k.
The notions of equivalent and reducible or irreducible parameterizations carry
over to the affine ones in the obvious way.
10.12.4. Theorem.
Given any parameterization we can always find a coordinate
system so that in that coordinate system the related affine parameterization is equiv-
alent to one of the form
()
=
()
=
m
Xt
t
m
m
Yt
at
+
a t
+
...,
1
2
1
2
where a
1
π 0, 0 < m, and 0 < m
1
< m
2
< ...
Proof.
See [Walk50].
The next theorem gives us a criterion for when a parameterization is reducible.
10.12.5. Theorem.
An affine parameterization of the form shown in Theorem
10.12.4 is reducible if and only if the integers m
1
, m
2
,...have a common factor larger
than 1.
Proof.
See [Walk50].
Definition.
A
place
of a plane curve is an equivalence class of irreducible parame-
terizations of the curve with respect to being equivalent. The point on the curve deter-
mined by the center of any one the representatives of a place is called the
center
of
the place.
A place is the algebraic version of what is called a branch of a function in complex
analysis and for that reason is sometimes also called a
branch
. Proposition 10.12.3
shows that the center of a place is well defined. One can talk about a place of a pro-
jective curve or any of its associated affine curves. One always talks about them in the
context of a particular representative for its equivalence class.
Proposition 10.12.1 proved that the center of a place is a point on the curve. At
this point, however, we do not know if any places or parameterizations even exist. To
put it another way, can curves be represented locally by power series? Therefore, the
next theorem is fundamental to the subject.
Every point of a plane curve in
C
2
10.12.6. Theorem.
is the center of at least one
but no more than a finite number of places.
Theorem 10.12.6 can be proved in different ways. We follow [Seid68] because this
approach will introduce certain local quadratic transformations which are useful for
computations. We start with a sequence of lemmas. Also, it will be convenient to work
with an affine representation of a curve in the rest of this section.
