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(1) Every rational function on
V
can be represented by a quotient p(X)/q(X),
where p(X), q(X) Œ k[X] and q(X) œ I(
V
).
(2) Two such quotients p
1
(X)/q
1
(X) and p
2
(X)/q
2
(X) represent the same rational
function on
V
if p
1
(X)q
2
(X) - p
2
(X)q
1
(X) ΠI(
V
).
Proof.
The fact that k(
V
) is well defined follows from the fact that k[
V
] is an inte-
gral domain by Theorem 10.13.2. The rest is an easy exercise.
Definition.
A rational function u on
V
is said to
regular at a point
a
in
V
and
a
is
called a
regular point
of u if it can be represented as a quotient of polynomial func-
tions p(X)/q(X), where q(
a
) π 0. In that case we call p(
a
)/q(
a
) the
value
of u at
a
and
denote it by u(
a
).
It follows from Proposition 10.13.8(2) that the value of a rational function at a
regular point is well defined.
10.13.9. Theorem.
Let k be an algebraically closed field. A rational function on an
irreducible affine variety
V
in k
n
that is regular at every point of
V
is a polynomial (or
regular) function.
Proof.
See [Shaf94].
Definition.
The set of regular points of a rational function is called its
domain
of
definition.
10.13.10. Proposition.
Let
V
be an irreducible affine variety.
(1) The domain of definition of a rational function on
V
is a nonempty open
subset.
(2) A rational function on
V
is completely specified by its values on any nonempty
open subset of its domain of definition.
(3) The intersection of the domain of definitions of a finite number of rational
functions on
V
is again a nonempty open subset of
V
.
Proof.
See [Shaf94].
Definition.
Let
V
Õ k
n
and
W
Õ k
m
be affine varieties with
V
irreducible. A
rational
function from
V
to
W
,
u:
VW
Æ
,
consists of an m-tuple u = (u
1
,u
2
,...,u
m
) of rational functions u
i
Πk(
V
) with the prop-
erty that if
a
is a regular point for all the u
i
, then
()
=
(
()
()
()
)
Œ
u
a
u
a
,
u
a
,...,
u
m
a
W
.
1
2
We call such a point
a
a
regular point
of u and u(
a
) is called the
image
of
a
. The set
of regular points of u is called the
domain
of definition of u and the set
