Graphics Reference
In-Depth Information
A set
X
in k
m
with parameterization
Input:
x
1
= f
1
(t
1
,t
2
, º , t
n
x
2
= f
2
(t
1
,t
2
, º
),
,t
n
),
x
m
= f
m
(t
1
,t
2
, º ,t
n
),
where f
i
Πk[t
1
, º ,t
n
]
Output:
A set of polynomials g
1
, g
2
, º , g
s
Πk[x
1
, º ,x
m
], so that the variety
V
= V(g
1
,g
2
,º ,g
s
) is the smallest variety in k
m
containing
X
.
Compute a Gröbner basis for the ideal I = <x
1
-f
1
,x
2
- f
2
, º , x
m
-f
m
> in
k[t
1
,º ,t
n
,x
1
,º x
m
] with respect to the lex order, where
...
< x
1
< t
n
< t
n
-
1
<
...
x
m
< x
m
-
1
<
< t
1
.
The elements g
j
of the Gröbner basis not involving the variables t
i
are the basis
of an ideal J and
V
= V(J).
Algorithm 10.11.1.
Implicitization algorithm using Gröbner bases.
Theorems 10.11.1 and 10.11.4 lead to Algorithm 10.11.1 for finding an implicit
representation for a parameterized set.
10.12
Places of a Curve
This section is concerned with studying the local structure of a plane curve. In
particular, we would like to analyze the curve in a neighborhood of a singularity.
The analysis will be carried out by expanding the curve locally using power
series. Again, if we were to jump right in and begin with all the relevant definitions,
readers new to algebraic geometry would probably find everything very abstract
and even if they would have no trouble following the technical details it would seem
to be a lot of formal mumbo-jumbo. For that reason, we shall start this section
with some motivation for what we are going to do. The motivation comes from
complex analysis. We will basically take a well-known theory in complex analysis
and translate it into an algebraic setting. Therefore, we start with a simple example
on the complex analysis side and subsequently show how its analysis has bearing on
the analysis of curves in algebraic geometry. This will hopefully clarify some of the
issues at stake. We are relying on the fact that every reader has had calculus and prob-
ably a little complex analysis, so that the ideas should make a little more sense here
and help the reader understand the more algebraic and abstract discussion that
follows.
