Nonlinear Equations - UFRJ

60 [CH. 5: REPRODUCING KERNEL SPACES
with the notation K i·(x, y) =
∂
∂x i
K(x, y), K·j (x, y) =
and K ij (x, y) =
∂ ∂
∂x i ∂ȳ j
K(x, y).
The Fubini 1-1 form is then:
√ −1 ∑
ω = g ij dz i ∧ d¯z j
2
and the volume element is 1 n!
∧ n
i=1 ω.
Exercise 5.2. Prove Lemma 5.10.
5.3 Root density
ij
∂
∂ȳ j
K(x, y)
We will deduce the famous theorems by Bézout, Kushnirenko and
Bernstein from the statement below. Recall that n K (f) is the number
of isolated zeros of f that belong to K.
Theorem 5.11 (Root density). root density Let K be a locally measurable
set of an n-dimensional manifold M. Let F 1 , . . . , F n be fewspaces.
Let ω 1 , . . . , ω n be the induced symplectic forms on M. Assume
that f = f 1 , . . . , f n is a zero average, unit variance variable in
F = F 1 × · · · × F n . Then,
E(n K (f)) = 1
π
∫K
n ω 1 ∧ · · · ∧ ω n .
Proof of Theorem 5.11. Let V ⊂ F×M, where F = F 1 ×F 2 ×· · ·×F n
be the incidence locus, V def
= {(f, x) : f(x) = 0}. (It is a variety when
M is a variety). Let π 1 : V → F and π 2 : V → M be the canonical
projections.
For each x ∈ M, denote by F x = {f ∈ F : f(x) = 0}. Then F x is
a linear space of codimension n in F. More explicitly,
F x = K 1 (·, x) ⊥ × · · · × K n (·, x) ⊥ ⊂ F 1 × · · · × F n
using the notation K i for the reproducing kernel associated to F i .
Let O ∈ M be an arbitrary particular point, and let F = F O .
We claim that (V, M, π 2 , F ) is a vector bundle.