Graphics Reference
In-Depth Information
mm
s
m
s
ggg
=
...
g
.
1
2
12
where the g
j
are irreducible and nonassociates and m
j
> 0. Since each f
i
is irreducible
and f
i
vanishes on V(g), it follows from Corollary 10.5.7 that f
i
divides g, that is, g =
f
i
h. Factoring h into irreducible polynomials and using the fact that we are in a unique
factorization domain, f
i
must be an associate of some g
j
. A similar argument shows
that each g
j
is an associate of some f
i
. These facts clearly imply the result.
Definition.
Let
S
be a hypersurface in
C
n
. By definition
S
= V(f) where f is a poly-
nomial of the form shown in equation (10.36). The polynomial f
1
f
2
...f
k
is called “the”
minimal polynomial
associated to V(f), its degree is called the
order
or
degree
of the
hypersurface
S
, and the equation
ff
...
f
k
=
0
12
is called “the”
minimal equation
for
S
. The degree of a hypersurface
S
will be denoted
by deg
S
.
In the context of higher dimensions the term “degree” is the one usually used, but
for curves, the term “order” seems to be the more common one. It suggests more that
we are talking about an invariant associated to a geometric set and not to a particu-
lar polynomial.
10.5.9. Corollary.
Two minimal polynomials associated to a hypersurface in
C
n
differ by a nonzero constant. Two minimal equations for such a hypersurface differ
by a nonzero constant.
Proof.
This is an easy consequence of Theorem 10.5.8.
It follows from Corollary 10.5.9 that minimal polynomials and equations for
hypersurfaces (over the complex numbers) are essentially unique and so in the fu-
ture we are justified in referring to “the” minimal polynomial or equation. They are
polynomials, respectively, equations of minimal degree. The order or degree of a
hypersurface is also well defined.
Definition.
A variety
V
is said to be
reducible
if it is the proper union of two other
varieties, that is,
V
=
V
1
»
V
2
, where
V
i
π
V
. Otherwise,
V
is said to be
irreducible
.
10.5.10. Example.
In
R
3
, the y-z plane V(X) and the x-z plane V(Y) are irreducible.
The variety V(XY) = V(X) » V(Y) is reducible.
10.5.11. Example.
The variety V(XZ,YZ) in
R
3
is reducible because it is the union
of two varieties, the z-axis and the xy-plane, that is, V(XZ,YZ) = V(Z) » V(X,Y).
The next example is less trivial.
Consider the varieties
V
= V(f
1
,f
2
) and
W
= V(f
1
,f
3
) in
R
3
, where
10.5.12. Example.
