Nonlinear Equations - UFRJ

[SEC. 10.4: THE GEOMETRIC VERSION... 153
The deterministic algorithm by Bürgisser and Cucker is similar,
with starting system
⎡ ⎤
ˆF 0 (X) =
⎢
⎣
X d1
1 − Xd1 0
X dn
n
.
− X d1
0
Therefore it is possible to average over all paths, because the starting
system is ‘symmetric’. The condition integral was bounded in two
parts. When t is small, the condition µ 2 (f t ) can be bounded in terms
of the condition of f 0 , which unfortunately grows exponentially in n.
The rest of the analysis relies on the following ‘smoothed analysis’
theorem:
Theorem 10.21. Let d = (d 1 , . . . , d n ), let ¯F ∈ H d and let F be
random with probability density N(¯F, σ 2 I). Then,
( ) µ
2
E 2 (F)
‖F‖ 2 ≤ n3/2
σ 2
I refer to the paper, but the reader may look at exercises 10.2
and 10.3 before.
Exercise 10.1. In Theorem 10.16, replace the variance by σ 2 . Show
(10.19).
Exercise 10.2. Show that the average over the complex ball B(0, ɛ) ⊂
C 2 of the function 1/(|z 1 | 2 + |z 2 | 2 ) is finite.
Exercise 10.3. Let n = 1 and d = 1. Then H d is the set of linear forms
in variables x 0 and x 1 . Compute the expected value of µ 2 2(f)/‖f‖.
Conclude that its expected value is finite, for F ∈ N(e 1 , σ).
10.4 The geometric version of Smale’s 17 th
problem
In view of Theorem 10.5, one would like to be able to produce given
F 1 ∈ H d , a path (f t , z t ) in the solution variety such that
⎥
⎦
1. An approximate zero X 0 is known for f 0 .