Nonlinear Equations - UFRJ

[SEC. 2.4: GROUP ACTION AND NORMALIZATION 23
In this situation, #Z(J) is not larger than the degree of A with
respect to k.
Example 2.18. n = 1, J = (x 2 ). In this case A = k 2 so r = 2. Note
however that #Z(J) = 1.
However, if we require J to be prime, the number of zeros is
precisely the degree [A : k]. The same principle holds for J =
(f 1 , . . . , f n ) for generic polynomials. We can prove now a version
of Bézout’s theorem:
Theorem 2.19 (Bézout’s Theorem, generic case). Let d 1 , . . . , d n ≥
1. Let B = d 1 d 2 · · · d n . Then generically, f ∈ P d1 × · · · × P dn has B
isolated zeros in k n .
Proof. Let J r = (f r+1 , . . . , f n ) and A r = k[x 1 , . . . , x n ]/J r .
Our induction hypothesis (in n − r) is:
[A r : k[x 1 , . . . , x r−1 ]] = d r+1 d r+2 . . . d n
When r = n, this is Proposition 2.11.
For r < n, A r is integral of degree d r over A r+1 . The integral
equation (in x r ) is, up to a multiplicative factor,
f r (x 1 , . . . , x r , y r+1 , . . . , y n ) = 0
where y r+1 , . . . , y n are elements of A r+1 (hence constants). Hence,
[A : k] = d 1 d 2 · · · d n .
Noether normalization provides information about the ring R =
k[x 1 , . . . , x n ].
Definition 2.20. A ring R is noetherian if and only if, there cannot
be an infinite ascending chain J 1 J 2 · · · of ideals in R.
Theorem 2.21. Let k be algebraically closed. Then R = k[x 1 , . . . ,
x n ] is Noetherian.