Geometric progressions in the sets of values of rational functions

Abstract

Let $a, Q\in\Q$ be given and consider the set $\mathcal{G}(a,\mathcal{Q})=\{a{\mathcal{Q}}^i:i\in \mathbb{N}\}$ of terms of geometric progression with 0th term equal to $a$ and the quotient $Q$. Let $f\in\Q(x, y)$ and ${\mathcal{V}}_f$ be the set of finite values of $f$. We consider the problem of existence of $a, Q\in\Q$ such that $\mathcal{G}(a,\mathcal{Q})\subset {\mathcal{V}}_f$. 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 $f_1,f_2\in \mathbb{Z}[x,y]$ are homogenous forms of degrees $d_1,d_2$ and $|d_1-d_2|=1$, we prove that $\mathcal{G}(a,\mathcal{Q})\subset {\mathcal{V}}_f$ 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.

17 pages