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Zeta functions. No.4
Yoshifumi Tsuchimoto
P
ROPOSITION
4.1
Let
be an irreducible polynomial in one variable of degree
. Let us consider
, an equation in one variable. Then:
D
EFINITION
4.2
Let
be a ring. A polynomial
is said to be
homogenius
of degree
if an equality
holds as a polynomial in
variables
.
D
EFINITION
4.3
Let
be a field.
We put
and call it (the set of
-valued points of) the
projective space
. The class of an element
in
is denoted by
.
Let
be homogenious polynomials. Then we set
and call it (the set of
-valued point of) the
projective variety
defined by
.
(Note that the condition
does not depend on the choice of the representative
of
.)
L
EMMA
4.4
We have the following picture of
.
That means,
is divided into two pieces
a nd
.
That means,
is covered by three ``open sets''
. Each of them is isomorphic to the plane (that is, the affine space of dimension 2).
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2013-05-10