Nonlinear Equations - UFRJ

138 [CH. 10: HOMOTOPY
Again, f t is the equivalence class of F t in P(H d ). Given representatives
for f 0 and f 1 , two cases arise: either we can find F 0 and F 1
satisfying (10.1) and (10.2), or we may find f 1/2 half-way in projective
space such that (f 0 , f 1/2 ) and (f 1/2 , f 1 ) fall into the previous case.
Therefore, (10.2) is not a big limitation.
Let 0 < a < α 0 , where α 0 is the constant of Theorem 9.9. We will
say that X is a (β, µ, a)-certified approximate zero of f if and only if
D 3/2
2 ‖X‖−1 β(F, X)µ(f, x) ≤ a.
This condition implies, in particular (Theorems 9.9 and 9.11) that X
is an approximate zero of the second kind for f.
We address the following computational task:
Problem 10.3 (true lifting). Given 0 ≠ F 0 and 0 ≠ F 1 ∈ H d
satisfying (10.1) and (10.2), and given also a (β, µ, a 0 )-certified approximate
zero X 0 of F 0 , associated to a root z 0 , find a (β, µ, a 0 )-
certified approximate zero of f 1 , associated to the zero z 1 where z t is
continuous and F t (z t ) ≡ 0 for t ∈ [0, 1].
A true lifting is not always possible. Moreover, the cost of the algorithm
will depends on certain invariant of the path (f t , z t ) that can
be infinite. However, we may understand this invariant geometrically.
The set V = {(f, z) ∈ P(H d ) × P n : f(z) = 0} is known as the
solution variety of the problem. The solution variety inherits a metric
from the product of the Fubini-Study metrics in P(H d ) and P n+1 .
The discriminant variety Σ ′ in V is the set of critical points for
the projection π 1 : V → H d . This is a Zariski closed set, hence its
complement is path-connected. For a probability one choice of F 0 , F 1 ,
the corresponding path (f t , z t ) exists and keeps a certain distance to
this discriminant variety. We will see that in that case, the algorithm
succeeds. Before we define the invariant:
Definition 10.4. The condition length of the path (f t , z t ) t∈[a,b] ∈ V
is
L(f t ; a, b) =
∫ b
a
µ(f s , z s )‖( f ˙ s , z˙
s )‖ (fs,z s) ds