Nonlinear Equations - UFRJ

150 [CH. 10: HOMOTOPY
Corollary 10.17. Let (f, z) ∈ V be random in the following sense:
f is normal with mean zero and variance σ 2 , and z is a random zero
of f (each one has same probability). Then,
( ) ( )
µ(f, z)
2 e
3/2
E
‖f‖ 2 2 − 1 nσ −2 .
Bürgisser and Cucker introduced the following invariant:
Definition 10.18.
µ 2 2 : P(H d ) → R,
f ↦→ 1 B
∑z∈Z(f)
µ(f, z)2
where B = ∏ d i is the Bézout number.
Define the line integral
∫ 1
M(f t ; 0, 1) =
0
‖ f ˙ t ‖ ft µ 2 2(f t )dt =
∫
µ 2 2(f t )dt.
(f t) t∈[0,1]
When F 1 is Gaussian random and F 0 , z 0 are random as above,
each zero z 0 of F 0 is equiprobable and
(∫ 1
)
E ‖ f ˙ t ‖ ft µ(f t , z t ) 2 dt = E (M(f t ; 0, 1))
0
Also, M(f t ; 0, 1) is a line integral in P(H d ), and depends upon F 0
and F 1 . The curve (f t ) t∈[0,1] is invariant under real rescaling of F 0
and F 1 .
Bürgisser and Cucker suggested to sample F 0 and F 1 in the probability
space (B(0, √ 2N), κ −1 dH d
)
instead of (H d , dH d ). Here, N is the complex dimension of sampling
space (H d and κ is the constant that makes the new sampling space
into a probability space. It is known that κ ≥ 1/2.
Therefore, when F 0 , Z 0 and F 1 are random in the sense of Proposition
10.15, the expected value of M will be computed as if F 0 , F 1
were sampled in the new probability space. We will need a geometric
lemma before proceeding.