Attila Pethő: On the diophantine equation with

Let be an algebraically closed field of characteristic 0, and . Let further and the sequence of polynomials be defined by the recursion

for all .

In this talk we are given a survey on results concerning the equation

 (1)

in integers with , where . We assume that are transcendental over , but algebraically dependent, i.e. holds. Our journey is based on joint works with Cl. Fuchs and R.F. Tichy as well as on a paper of U. Zannier. If then (1) has up to two exceptional families only finitely many solutions and Zannier proved a quite good bound for the number of solutions. In the general case we have also a quite satisfactory description of the exceptional cases.

We finish our talk by some open problem.