Nonlinear Equations - UFRJ

70 [CH. 5: REPRODUCING KERNEL SPACES
Corollary 3.9 completes the proof.
We can prove Bézout’s Theorem by combining Theorem 5.21 with
Corollary 5.12. The multi-homogeneous Bézout theorem is more intricate
and implies Bézout’s theorem, so we write down a formal proof
of it instead.
Proof of Theorem 1.5. Let H = (C × ) s act in C n+s \ V −1 (0) as explained
above. Then H d1 , . . . , H dn are fewspaces of equations on
C n+s /H = P n1 × · · · × P ns
which is compact. The group U(n 1 + 1) × · · · × U(n s + 1) acts transitively
and preserves the symplectic forms.
It remains to prove that the set of critical points of π 1 is contained
in a Zariski closed set. We proceed by induction in s.
The case s = 1 (Bézout’s theorem setting) follows directly from
the Main Theorem of Elimination Theory (Th.2.33) applied to the
systems f 1 (x) = 0, · · · , f n (x) = 0, g j (x) = 0 where g(x) is the determinant
of Df(x) e ⊥
j
. According to that theorem, Σ j = {f : ∃x ∈ P n :
f 1 (x) = · · · = f n (x) = g j (x) = 0} is Zariski closed. Hence Σ = ∩Σ j
is Zariski closed.
For the induction step, we assume that the induction hypothesis
above was established up to stage s − 1. As before,
Σ j = {(f, x 1 , . . . , x s−1 ) : ∃x ∈ P ns : f 1 (x) = · · · = f n (x) = g J (x) = 0}
with g J (x s ) = det Df(x) J and J is a coordinate space of C n+s of
dimension n. By Theorem 2.33 Σ ′ = ∩Σ ′ J is a Zariski closed subset
of F × C n1+···+ns−1+s−1 . Its defining polynomial(s) are homogeneous
in x 1 , . . . , x s . Then by induction, we know that the set Σ of all f
such that those defining polynomials vanish for some x 1 , . . . , x s−1 is
Zariski closed.
As it is a zero-measure set, Σ F. Thus, the set F \ Σ of regular
values of π 1 is path-connected. Theorem 1.5 is now a consequence of
Theorem 5.21 together with Corollary 5.14.