Graphics Reference
In-Depth Information
(
) = 0
fXY
,
.
Assume that p 0 = (x 0 ,y 0 ) lies on C . The line L through p 0 with slope t has equation
(
)
Yy tXx
=+ -
0 .
(10.82)
0
We need to solve
(
(
)
) =
fXy
,
+-
tX x
0
(10.83)
0
0
for X. Rather than using the standard quadratic formula which would seem to involve
square roots, note that we already have one solution x 0 . It follows that the second root
x 1 satisfies
xx
+=-,
B
1
where B is the coefficient of X in equation (10.83) after one has divided the equation
by the coefficient of the X 2 term. Therefore, the second intersection (x 1 ,y 1 ) of L with
C is determined by rational functions in t since one can use equation (10.82) to solve
for y 1 .
Because conics can be described by rational functions, they are called “rational”
curves. Before we introduce the terminology required to study rational functions, let
us look at the simpler case of polynomial functions.
Let V Õ k n and W Õ k m . A function
Definition.
u: VW
Æ
is called a polynomial function from V to W if there exist polynomials p 1 , p 2 ,..., p m
Πk[X 1 ,X 2 ,...,X n ] such that
n
() =
(
()
()
()
)
u
a
p
a
,
p
a
,...,
p
a
,
for all
a
Œ
k
.
1
2
m
The polynomials p i are called representatives for the function u. If m = 1 and W = k,
then u will be called simply a polynomial or regular function on V . The set of polyno-
mial functions
u
: V Æ
k
on V will be denoted by k[ V ].
The representatives p i of a polynomial function are typically not unique. For
example, if V is a hypersurface V(f), then p i and p i + f define the same function on V
because f vanishes on V . Note that pointwise addition and multiplication makes k[ V ]
into a ring.
Let V Õ k n .
10.13.2. Theorem.
[
]
kX X
,
,...,
X
12
n
(1) k[ V ] is isomorphic to
.
()
I
V
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