Graphics Reference
In-Depth Information
One can define a formal derivative for power series and polynomials.
Definition.
Let
()
=
2
i
fX
a
+
aX a X
+
+ ◊◊◊+
aX
+ ◊◊◊
0
1
2
i
be a formal power series (or polynomial) over a ring R with unity. Define the
deriva-
tive of f
, denoted by f ¢(X) or df/dX, by
2
i
-
1
fX a
¢
()
=
+
2
aX
+
3
aX
+ ◊◊◊+
iaX
i
+ ◊◊◊
.
1
2
3
If f(X
1
,X
2
,...,X
n
) is a multivariate formal power series (or polynomial), define the
ith
partial derivative of f
, denoted by ∂f/∂X
i
, to be the derivative of f thought of as a formal
power series (or polynomial) in X
i
.
It is easy to see that this definition of derivative or partial derivative of a polyno-
mial matches the corresponding definition that one encounters in calculus if one
thinks of the polynomials as functions. It also satisfies the same properties. The main
point is that there is no need to introduce the notion of limits and the purely alge-
braic aspects of the derivative turn out to have important applications in algebra, in
particular, algebraic geometry.
B.7.8. Theorem.
(Euler) If f(X
1
,X
2
,...,X
n
) is a homogeneous polynomial of degree
d in the variables X
i
, then
n
∂
Â
(
)
=
(
)
X
f X
,
X
,...,
X
df X
,
X
,...,
X
.
i
12
n
12
n
∂
X
i
i
=
1
Proof.
Clearly, it suffices to prove the result in the case where f is a monomial
dd
d
XX
1
2
◊◊◊
X
,
d d
+
+◊◊◊+
d
=
d
.
n
1
2
n
12
But in that case,
∂
(
)
=
(
)
X
fX X
,
,...,
X
dfX X
,
,...,
X
,
i
12
n
i
12
n
∂
X
i
from which the theorem easily follows.
B.7.9. Theorem.
(Hilbert Basis Theorem) If R is a Noetherian ring, then so is R[X].
Proof.
See [Jaco66].
B.7.10. Corollary.
If R is a Noetherian ring, then so is R[X
1
,X
2
,...,X
n
].
Proof.
This uses Theorem B.7.9 and induction.
B.7.11. Theorem.
If R is a Noetherian ring, then so is R[[X
1
,X
2
,...,X
n
]].
Proof.
See [ZarS60], Volume II.
