In 1965, Birch, Chowla, Hall, and Schinzel posed a problem about the possible minimum degree of the difference where and are two coprime polynomials with complex coefficients. The above problem was generalized by Zannier in 1995 as follows: let and be two coprime polynomials of degree having the following factorization patterns:
In this expressions the multiplicities and are given, while the roots and are not fixed, though they must all be distinct. The problem is to find the minimum possible degree of the difference Zannier proved that
and this bound is always attained.
The triples for which this bound is attained are called Davenport-Zannier triples. Davenport-Zannier triples defined over are the most interesting ones since by specializing to a rational value one may obtain an important information concerning differences of integers with given factorization patterns. In the talk based on a recent joint paper with A. Zvonkin we relate the problem of description of Davenport-Zannier triples defined over with the Grothendieck theory of "Dessins d'enfants" and present a method which permits to produce "most" of such triples.