Geometric progressions in the sets of values of rational functions (Q6585091)

Let \(a, Q\in \mathbb{Q}\) be given and consider the set \(\mathcal{G}(a, Q)=\{aQ^{i}:\;i\in \mathbb{N}\}\) of terms of geometric progression with 0th term equal to \(a\) and the quotient \(Q\). Let \(f\in \mathbb{Q}(x, y)\) and \(\mathcal{V}_{f}\) be the set of finite values of \(f\). The author considered the problem of existence of \(a, Q\in \mathbb{Q}\) such that \(\mathcal{G}(a, Q)\subset\mathcal{V}_{f}\).\N\NThe author proves that for \(f \in Q(x)\) and any \(a, Q \in \mathbb{Q}\), the equation \(f(x)=aQ^i\) has only finitely many solutions in \(x \in \mathbb{Q}\) and \(i \in \mathbb{N}\) (see Theorem 2.1). The author describes several classes of rational function for which the above problem has a positive solution (see Theorems 2.2-2.4, Example 2.6, Theorem 2.8). 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 homogeneous forms of degrees \(d_{1}, d_{2}\) and \(|d_{1}-d_{2}|=1\), he proved that \(\mathcal{G}(a, Q)\subset \mathcal{V}_{f}\) if and only if there are \(u, v\in \mathbb{Q}\) such that \(a=f(u, v)\).\N\NThe author also studies the stated problem for the rational function \(f(x, y)=(y^2-x^3)/x\). He relates the problem to the existence of rational points on certain elliptic curves (see Theorem 3.1). As a consequence of his investigations, he proves that for each \(a\in\{1, \ldots, 10^{3}\}\) such that the elliptic curve \(y^2=x^3+ax\) has positive rank, there are infinitely many values of \(Q\in \mathbb{Q}\) such that \(Q\) is not a square and \(\mathcal{G}(a, Q)\subset \mathcal{V}_{f}\).\N\NAt last, the author presents interesting numerical observations which allow him to state several questions and conjectures.