Nonlinear Equations - UFRJ

[SEC. 6.3: GEOMETRIC CONSIDERATIONS 77
Thus,
ˆm ∗ η(u, v) = D 2 ( 1 log K(x, x))(re(u), im(v))
2
using Lemma 5.10.
−D 2 ( 1 log K(x, x))(im(u), re(v))
2
= 2 n 〈u, Jv〉 x+
√ −1y
= 2 n ω x+
√ −1y (u, v)
As a consequence toward the proof of Kushnirenko’s theorem, we
note that
Proposition 6.8.
E(n M (f)) = n!Vol(A)
Theo-
Proof. The preimage M = m −1 (A) has volume π n Vol(A).
rem 5.11) implies then that expected number of roots is
E(n M (f)) = 1
π n ∫M
n∧
i=1
ω = n! Vol(M) = n!Vol(A).
πn 6.3 Geometric considerations
To achieve the proof of the Kushnirenko theorem, we still need to
prove that the number of roots is generically constant. The following
step in the proof of that fact was used implicitly in other occasions:
Lemma 6.9. Let M be a holomorphic manifold, and F = F 1 ×· · ·×F n
be a product of fewspaces. Let V ⊂ F × M and let π 1 : V → F and
π 2 : V → M be the canonical projections.
Assume that (f t ) t∈[0,1] is a smooth path in F and that for all t, f t
is a regular value for f t . Let v 0 ∈ π1 −1 (f 0).
Then, the path f t can be lifted to a path v t with π 1 (v t ) = f t in an
interval I such that either I = [0, 1] or I = [0, τ), τ < 1 and π 2 (v t )
diverges for t → τ.