Nonlinear Equations - UFRJ

162 [CH. A: OPEN PROBLEMS
is polynomial for, say, sparse polynomial systems. Same question is
open for real polynomial systems (i.e. for polynomial systems in H d
with real coefficients). Some progress in this last problem has been
done in [22]. Another interesting question is if some of these methods
can be made to work for polynomial systems given by straight line
programs.
A.6 Numerics for decision problems
Most of the algorithms nowadays used for polynomial system solving
are based on numerics, for example all the homotopy methods
discussed above. However, many problems in computation are decissional
problems. The model problem is Hilbert’s Nullstellensatz,
that is given f 1 , . . . , f k polynomials with unknowns z 1 , . . . , z n , does
there exist a common zero ζ ∈ C n ? This problem asks if numeric
algorithms can be designed to answer this kind of questions. Note
that Hilbert’s Nullstellensatz is a NP –hard problem, so one cannot
expect worse case polynomial running time, but maybe average polynomial
running time can be reached. Some progress in this direction
may be available using the algorithms and theorems in [13, 25].
A.7 Integer zeros of a polynomial of one
variable
A nice problem to include in this list is the so–called Tau Conjecture:
is the number of integer zeros of a univariate polynomial, polynomially
bounded by the length of the straight line program that generates
it? This is Smale’s 4th problem and we refer the reader to [79].
Another problem is the following: given f 1 , . . . , f k integer polynomials
of one variable, find a bound for the maximum number of
distinct integer roots of the composition f 1 ◦ · · · ◦ f k . In particular,
can it happen that this number of zeros is equal to the product of
the degrees?
This problem has been studied by Carlos Di Fiori, and he found
an example of 4 polynomials of degree 2 such that their composition