Nonlinear Equations - UFRJ

[SEC. 5.3: ROOT DENSITY 61
First, we should check that V is a manifold. Indeed, V is defined
implicitly as ev −1 (0), where ev(f, x) = f(x) is the evaluation function.
Let p = (f, x) ∈ V be given. The differential of the evaluation
function at p is
Dev(p) : ḟ, ẋ ↦→ Df(x)ẋ + ḟ(x).
Let us prove that Dev(p) has rank n.
⎡
〈 ˙
⎤
f 1 (·), K 1 (·, x)〉 F1
Dev(p)(ḟ, 0) = ⎢
⎥
⎣ . ⎦
〈 f ˙ n (·), K n (·, x)〉 Fn
and in particular, Dev(p)(e i K i (x, ·)/K i (x, x), 0) = e i . Therefore 0 is
a regular value of ev and hence (Proposition 3.2) V is an embedded
manifold.
Now, we should produce a local trivialization. Let U be a neighborhood
of x. Let i O : F x → F be a linear isomorphism. For y ∈ U,
we define i y : F y → F x by othogonal projection in each component.
The neighborhood U should be chosen so that i y is always a linear
isomorphism. Explicitly,
1
i y = I F1 −
K 1 (x, x) K 1(x, ·)K 1 (x, ·) ∗ ⊕ · · ·
⊕ I Fn −
so U = {y : K j (y, x) ≠ 0 ∀j}.
For q = (g, y) ∈ π2 −1 (x), set
This is clearly a diffeomorphism.
1
K n (x, x) K n(x, ·)K n (x, ·) ∗
Φ(q) = (π 2 (q), i O ◦ i y ◦ π 1 (q)).
The expected number of roots of F is
∫
E(n K (f)) = χ π
−1
2 (K)(p)(π∗ 1dF)(p).
V