Nonlinear Equations - UFRJ

[SEC. 1.4: SMALE’S 17 TH PROBLEM 11
1.4 Smale’s 17 th problem
Theorems like Bézout’s or Bernstein’s give precise information on the
solution of systems of polynomial equations. Proofs of those theorems
(such as in Chapters 2, 5 or 6) give a hint on how to find those roots.
They do not necessarily help us to find those roots in an efficient way.
In this aspect, nonlinear equation solving is radically different
from the subject of linear equation solving, where algorithms have
running time typically bounded by a small degree polynomial on the
input size. Here the number of roots is already exponential, and even
finding one root can be a desperate task.
As in numerical linear algebra, nonlinear systems of equations
may have solutions that are extremely sensitive to the value of the
coefficients. Instances with such behavior are said to be poorly conditioned,
and their ‘hardness’ is measured by an invariant known as the
condition number. It is known that the condition number of random
polynomial systems is small with high probability (See Chapter 8).
Smale 17 th problem was introduced in [78] as:
Open Problem 1.11 (Smale). Can a zero of n complex polynomial
equations in n unknowns be found approximately , on the average, in
polynomial time with a uniform algorithm?
The precise probability space referred in [78] is what we call
(H d , dH d ) in Chapter 5. Zero means a zero in projective space P n ,
and the notion of approximate zero is discussed in Chapter 7. Polynomial
time means that the running time of the algorithm should
be bound by a polynomial in the input size, that we can take as
N = dim H d . The precise model of computation will not be discussed
in this book, and we refer to [20]. However, the algorithm should be
uniform in the sense that the same algorithm should work for all
inputs. The number n of variables and degrees d = (d 1 , . . . , d n ) are
part of the input.
Exercise 1.9. Show that N = ∑ ( )
n di + n
i=1
. Conclude that there
n
cannot exist an algorithm that approximates all the roots of a random
homogeneous polynomial system in polynomial time.