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needs the points “at infinity.” To do this one must express curves in homogeneous
coordinates and via homogeneous polynomials. This means that it is important that
one can relate properties of polynomials with their homogenized versions and vice
versa. The next two results basically say that the answers to questions about factor-
ization of polynomials for varieties are the same for projective varieties and their affine
counterparts and so we may permit ourselves to not explicitly state which type of
variety we are talking about in such cases.
10.5.1. Proposition.
Any factor of a homogeneous polynomial is homogeneous.
Proof.
Exercise.
10.5.2. Theorem.
Let F be a homogeneous polynomial and f =
D(F) its
dehomogenization.
(1) Each factor of F dehomogenizes to a factor of f and conversely each factor of
f homogenizes to a factor of F.
(2) F is irreducible if and only if f is.
Proof.
Exercise.
10.5.3. Proposition.
Let k be an algebraically closed field and let f(X,Y) be a homo-
geneous polynomial of degree d in k[X,Y]. Then f can be factored into linear factors,
that is, f has the form
d
'
1
(
)
=
(
)
fXY
,
aY bX
-
,
a b
,
Œ
k
.
i
i
i
i
i
=
Proof.
Write f in the form
X g
Y
X
Ê
Ë
ˆ
¯
(
)
=
d
fXY
,
,
where g(Z) is a polynomial of degree d in k[Z]. Since k is algebraically closed, g (as a
polynomial in Z) factors into linear factors. Replacing Z by Y/X in this factorization
and simplifying gives the result.
For example,
2
È
˘
Y
X
Y
X
Ê
Ë
ˆ
¯
2
2
2
X
-
3
YYX
+
2
=
1
-
3
+
2
Í
˙
Î
˚
Y
X
YX YX
Y
X
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
2
=
X
-
12
-
1
(
)
(
)
.
=-
2
-
Note that we could equally well have put things in terms of X/Y rather than Y/X,
because
