Geometric progressions in the sets of values of rational functions

Abstract

Let $a, Q\in\Q$ be given and consider the set $\cal{G}(a, Q)=\{aQ^{i}:\;i\in\N\}$ of terms of geometric progression with 0th term equal to $a$ and the quotient $Q$. Let $f\in\Q(x, y)$ and $\cal{V}_{f}$ be the set of finite values of $f$. We consider the problem of existence of $a, Q\in\Q$ such that $\cal{G}(a, Q)\subset\cal{V}_{f}$. In the first part of the paper we describe several classes of rational function for which our problem has a positive solution. In particular, if $f(x,y)=\frac{f_{1}(x,y)}{f_{2}(x,y)}$, where $f_{1}, f_{2}\in\Z[x,y]$ are homogenous forms of degrees $d_{1}, d_{2}$ and $|d_{1}-d_{2}|=1$, we prove that $\cal{G}(a, Q)\subset \cal{V}_{f}$ if and only if there are $u, v\in\Q$ such that $a=f(u, v)$. In the second, experimental, part of the paper we study the stated problem for the rational function $f(x, y)=(y^2-x^3)/x$. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures.