In 1965, Birch, Chowla, Hall, and Schinzel posed a problem about the possible minimum degree of the difference $R=A^3-B^2,$ where $A$ and $B$ are two coprime polynomials with complex coefficients. The above problem was generalized by Zannier in 1995 as follows: let $P$ and $Q$ be two coprime polynomials of degree $n$ having the following factorization patterns:

\begin{displaymath}P(x)=\prod_{i=1}^p(x-a_i)^{\alpha_i},    Q(x)=\prod_{j=1}^q(x-b_j)^{\beta_j}.\end{displaymath}

In this expressions the multiplicities $\alpha_i$ and $\beta_j$ are given, while the roots $a_i$ and $b_j$ are not fixed, though they must all be distinct. The problem is to find the minimum possible degree of the difference $R=P-Q.$ Zannier proved that

\begin{displaymath}\deg R\geq (n+1)-(p+q),\end{displaymath}

and this bound is always attained.

The triples $(P,Q,R)$ for which this bound is attained are called Davenport-Zannier triples. Davenport-Zannier triples defined over $\mathbb{Q}$ are the most interesting ones since by specializing $x$ 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 $Q$ with the Grothendieck theory of "Dessins d'enfants" and present a method which permits to produce "most" of such triples.