Combinatorial Computations on an Extension of a Problem by Pál Turán

Abstract

Turan’s problem asks what is the maximal distance from a
polynomial to the set of all irreducible polynomials over Z.
It turns out it is sufficient to consider the problem in the setting of F2.

Even though it is conjectured that there exists an absolute constant C such that
the distance L(f - g) <= C, the problem remains open. Thus it attracts different
approaches, one of which belongs to Lee, Ruskey and Williams, who study
what the probability is for a set of polynomials ‘resembling’ the irreducibles
to satisfy this conjecture. In the following article we strive to provide more
precision and detail to their method, and propose a table with better numeric
results.

ACM Computing Classification System (1998): H.1.1.

*This author is partially supported by the High School Students Institute of
Mathematics and Informatics.