Geometric progressions in the sets of values of rational functions

Pub Date : 2024-07-01 DOI:10.1016/j.indag.2023.08.005

Maciej Ulas

引用次数: 0

Abstract

Let $a, Q \in Q$ be given and consider the set $G (a, Q) = {a Q^{i} : i \in N}$ of terms of geometric progression with 0th term equal to $a$ and the quotient $Q$ . Let $f \in Q (x, y)$ and $V_{f}$ be the set of finite values of $f$ . We consider the problem of existence of $a, Q \in Q$ such that $G (a, Q) \subset V_{f}$ . In the first part of the paper we describe certain classes of rational functions for which our problem has a positive solution. 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.

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有理函数值集中的几何级数

设 a,Q∈Q，并考虑第 0 项等于 a 的几何级数的项集 G(a,Q)={aQi:i∈N}，以及商 Q。设 f∈Q(x,y)，Vf 为 f 的有限值集。在论文的第一部分，我们描述了我们的问题有正解的几类有理函数。在论文的第二部分，即实验部分，我们研究了有理函数 f(x,y)=(y2-x3)/x 的既定问题。我们将这一问题与某些椭圆曲线上有理点的存在联系起来，并提出了有趣的数值观察结果，从而提出了几个问题和猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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