Nonlinear Equations - UFRJ

[SEC. A.3: EQUIDISTRIBUTION OF ROOTS UNDER UNITARY TRANSFORMATIONS159
in a much faster way than this proved in the Bezout Series (see (A.1)
below). With this new method, the Average Las Vegas algorithm
was improved to have running time which is almost quadratic in the
input size, see [13]. Not only the expected value of the running time
is known to be polynomial in the size of the input, also the variance
and other higher moments, see [16].
The existence of a deterministic polynomial time algorithm for
Smale’s 17th problem is still an open problem. In [25] a deterministic
algorithm is shown that has running time O(N log log N ), and indeed
polynomial time for certain choices of the number of variables and
degree of the polynomials. There exists a conjecture open since the
nineties [74]: the number of steps will be polynomial time on the
average if the starting point is the homogeneization of the identity
map, that is
⎧
z ⎪⎨
d1−1
0 z 1 = 0
f 0 (z) = .
, ζ 0 = (1, 0, . . . , 0).
⎪⎩
z dn−1
0 z n = 0
Another approach to the question is the one suggested by a conjecture
in [15] on the averaging function for polynomial system solving.
A.3 Equidistribution of roots under unitary
transformations
In the series of articles mentioned in the Smale’s 17th problem section,
all the algorithms cited use linear homotopy methods for solving
polynomial equations. That is, let f 1 be a (homogeneous) system to
be solved and let f 0 be another (homogeneous) system which has
a known (projective) root ζ 0 . Let f t be the segment from f 0 to f 1
(sometimes we take the projection of the segment onto the set of systems
of norm equal to 1). Then, try to (closely) follow the homotopy
path, that is the path ζ t such that ζ t is a zero of f t for 0 ≤ t ≤ 1.
If this path does not have a singular root, then it is well–defined. A
natural question is the following: Fix f 1 and consider the orbit of f 0
under the action f 0 ↦→ f 0 ◦ U ∗ where U is a unitary matrix. The root