Nonlinear Equations - UFRJ

136 [CH. 10: HOMOTOPY
form
for
x i+1 = N proj (F ti , x i ),
F t = (1 − t)F 0 + tF 1 , 0 = t 0 ≤ t i ≤ t τ = 1.
The major difficulty was finding an adequate starting pair (F 0 , x 0 ).
Only the existence of such a pair was known, without any clue on how
to find one in polynomial time.
A minor difficulty was the choice of the t i . This can be done
by trial and error. By doing so, there is no guarantee that one is
approximating an actual continuous solution path F t (x t ) ≡ 0. This is
trouble when attempting to find all the roots of a polynomial system,
or when investigating the corresponding Galois group.
In 2006, Carlos Beltrán and Luis Miguel Pardo demonstrated in
his doctoral thesis [6, 11] the existence of a good ‘questor set’ from
which an adequate random pair (F 0 , x 0 ) could be drawn with a good
probability.
A randomized algorithm is said to be of Vegas type if it returns an
answer with probability 1 − ɛ for some ɛ, and the answer it returns is
always correct. This is by opposition to Monte-Carlo type algorithms,
that would return a correct answer with probability 1 − ɛ.
Theorem 10.2 (Beltrán and Pardo). Let ɛ > 0. Then there is a
Vegas type algorithm such that, given n, d = d 1 , . . . d n and a random
F 1 ∈ (H d , dH d ), finds with probability 1 − ɛ an approximate zero X
for F 1 , within expected time O(N 5 ɛ −2 ), where N = dim H d is the
input size.
This result and its proof was greatly improved in subsequent papers
by Beltrán and Pardo such as [13]. The running time was reduced
to
E(τ) = C(max d i ) 3/2 nN
homotopy steps.
In another development, Peter Bürgisser and Felipe Cucker gave a
deterministic algorithm for solving random systems within expected
O(log log N)
E(τ) = N