Attila Pethő: On the diophantine equation $ G_n(x) = G_m(y)$ with $ Q(x,y) = 0$

Let $ {\bf K}$ be an algebraically closed field of characteristic 0, $ P(X) \in {\bf K}[X]$ and $ Q(X,Y) \in {\bf K}[X,Y]$. Let further $ A_0,\dots,A_{d-1},G_0,\dots,G_{d-1} \in {\bf K}[X]$ and the sequence of polynomials $ \{G_n(X)\}$ be defined by the recursion

$\displaystyle G_{n+d}(X)=A_{d-1}(X)G_{n+d-1}(X)+\ldots+A_0(X)G_{n}(X)

for all $ n\ge 0$.

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

$\displaystyle G_n(x) = c G_m(y)$ (1)

in integers $ n,m$ with $ n\not= m$, where $ c=c(m,n)\in {\bf K}^*$. We assume that $ x,y$ are transcendental over $ {\bf K}$, but algebraically dependent, i.e. $ Q(x,y) = 0$ 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 $ Q(X,Y)=Y-P(X)$ 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.

Back to the Index

Please send comments and corrections to Thomas Klausner.