Nonlinear Equations - UFRJ

[SEC. 10.1: HOMOTOPY ALGORITHM 137
homotopy steps. They pointed out that this solves Smale’s 17 th problem
for the ‘case’ max d i ≤ n 1
1+ɛ while the ‘case’ max d i ≥ n 1+ɛ
follows from resultant based algorithms such as [67]. When
Smale’s 17 th problem is still open.
n 1
1+ɛ ≤ max di ≤ n 1+ɛ ,
Another recent advance are ‘condition-length’ based algorithms.
While previous algorithm have a complexity bound in terms of the
line integral of µ(F t , z t ) 2 in P(H d ), condition-length algorithms (suggested
in [14,69] and developed in [7,31] have a complexity bound in
terms of a geometric invariant, the condition length. This allows to
reduce Smale’s 17 th problem (Open Problem 1.11) to a ‘variational’
problem.
In the rest of this chapter, I will give a simplified version of the
algorithm in [31], together with its complexity analysis. Then, I will
discuss how to use this algorithm to obtain results analogous to those
of [13] and [25]. In the last section, I will review some recent results
on the geometry of the condition metric.
10.1 Homotopy algorithm
Let d = (d 1 , . . . , d n ) be fixed, and set D = max d i . Recall that
H d is the space of homogeneous polynomial systems in n variables of
degree d 1 , . . . , d n . We want to find solutions z ∈ P n , and those will be
represented by elements of C n+1 \{0}. We keep the convention of the
previous chapter, where we set Z for a representative of z. However,
we will prefer representatives with norm one whenever possible.
We will consider an affine path in H d given by
F t = (1 − t)F 0 + tF 1
where F 0 and F 1 are scaled such that
‖F 0 ‖ = 1 F 0 ⊥ F 1 − F 0 (10.1)
with an extra bound,
‖F 1 − F 0 ‖ ≤ 1. (10.2)