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sible in the world of continuous functions and convergent series, but we should not
expect this representation to be unique. One reason for the nonuniqueness can be
explained by the that fact that the curve is derived from a projective curve that can
be coordinatized in many ways. For this and other reasons it is better to start with a
projective curve from the beginning. Our discussion will follow the presentation of
places of curves as given in [Walk50].
Let
C
be a plane curve in
P
2
(k). As usual, because it is convenient to study the
curve in the context of a coordinate system, we pick one, but one of our tasks as we
go along is to make sure that everything we do is independent of such a choice.
Assume that the curve
C
is defined by an equation
(
)
= 0
FXYZ
,,
,
where F(X,Y,Z) a homogeneous polynomial in our coordinate system.
Definition.
A (
projective
)
parameterization
of
C
(with respect to the given coordinate
system) is a triple (X(t),Y(t),Z(t)) of rational functions X(t), Y(t), Z(t) Πk((t)) satisfying
(1) F (X(t),Y(t),Z(t)) = 0.
(2) For no h(t) Πk((t)) do all of the products h(t)X(t), h(t)Y(t), or h(t)Z(t) belong
to k.
From a technical point of view, to make sense of some statements we should think
of our curve as a curve in
P
2
(k((t))), the projective plane over the field k((t)). For
example, condition (1) only really makes sense in that context, namely, we think of F
as a polynomial over k((t)). We shall not emphasize this point to keep the discussion
simple. All the important concepts will live in
P
2
(k). Also, condition (2) can be
rephrased as saying that [X(t),Y(t),Z(t)] is a point of
P
2
(k((t))) that does not lie in
P
2
(k). It ensures that a parameterization deals with nonconstant, nontrivial rational
functions. We could pick representatives X(t), Y(t), and Z(t) that are formal power
series, but this will not always be possible when we switch to affine coordinates. The
field k((t)) is the algebraic analog of the meromorphic functions in analysis. What we
call a parameterization here is sometimes called a
branch representation
(see [Seid68]).
Given how coordinate transformations are defined, we leave it to the reader to
check that a parameterization defined in one coordinate system will be a parameter-
ization (satisfying (1) and (2)) when transformed to another coordinate system (Exer-
cise 10.12.1). This means that a parameterization defined in one coordinate system
defines a well-defined parameterization in any other coordinate system. We shall allow
ourselves to talk about “the parameterization (X(t),Y(t),Z(t))” even though strictly
speaking we shall mean [X(t),Y(t),Z(t)]. Furthermore, since it is the equivalence class
[X(t),Y(t),Z(t)] that is important, we shall feel free to switch to whatever representa-
tive (X(t),Y(t),Z(t)) is convenient.
Let (X(t),Y(t),Z(t)) be a parameterization of
C
and let
(
(
()
)
(
( )
)
(
( )
)
)
m
=-
min
ord X t
,
ord Y t
,
ord Z t
.
By multiplying each coordinate by t
m
we may assume that our parameterization has
the property that each coordinate is a formal power series
