(G,F)-points on ℚ-algebraic varieties

Yangcheng Li and Hongjian Li
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 747–760
DOI: 10.7546/nntdm.2025.31.4.747-760
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Authors and affiliations

Yangcheng Li
School of Mathematical Sciences, South China Normal University
Guangzhou 510631, Guangdong, P. R. China

Hongjian Li
School of Mathematics and Statistics, Guangdong University of Foreign Studies
Guangzhou 510006, Guangdong, P. R. China

Abstract

Let \mathbb{Q} be the field of rational numbers, and let C be an algebraically closed field containing \mathbb{Q}. Let G\in \mathbb{Q}[x,y,z] be a polynomial, then the zero set of G is Z(G)=\{P\in C^n \mid G(P)=0\}. A set V\subset C^n is called a \mathbb{Q}-algebraic variety if V = Z(G) for some polynomial G in \mathbb{Q}[x,y,z]. The set V(G)=\{P\in\mathbb{Q}^3~|~G(P)=0\} is called the set of \mathbb{Q}-rational points of V. Let

    \[\begin{split} F:\quad &\mathbb{Q}^3\rightarrow \mathbb{Q}^3,\\ &(x,y,z)\mapsto (f(x),f(y),f(z)) \end{split}\]

be a vector function, where f\in \mathbb{Q}[x]. It is easy to show that the function obtained by the composition of G and F, denoted as G\circ F, is still in \mathbb{Q}[x,y,z]. Moreover, let V(G\circ F) be the set of \mathbb{Q}-rational points of the \mathbb{Q}-algebraic variety corresponding to G\circ F, i.e., V(G\circ F)=\{P\in\mathbb{Q}^3~|~G\circ F(P)=0\}. A rational point P is called a (G,F)-point on V(G) if P belongs to the intersection of V(G) and V(G\circ F), that is P\in V(G)\cap V(G\circ F). Denote \langle G,F\rangle as the set consisting of all (G,F)-points on V(G). Obviously, \langle G,F\rangle is the set of \mathbb{Q}-rational points of a \mathbb{Q}-algebraic variety, that is, \langle G,F\rangle=\{P\in\mathbb{Q}^3~|~G(P)=0~\text{and}~G\circ F(P)=0\}. In this paper, we consider the algebraic variety \langle G,F\rangle for some specific functions G and F. For these specific functions G and F, we prove that \langle G,F\rangle will be isomorphic to a certain elliptic curve. We also analyze some properties of these elliptic curves.

Keywords

  • Algebraic varieties
  • Elliptic curve
  • Diophantine equation
  • Rational solutions

2020 Mathematics Subject Classification

  • 11D25
  • 11D72
  • 14A10
  • 11G05

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Manuscript history

  • Received: 4 May 2025
  • Accepted: 26 October 2025
  • Online First: 28 October 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Li, Y., & Li, H. (2025). (G,F)-points on ℚ-algebraic varieties. Notes on Number Theory and Discrete Mathematics, 31(4), 747-760, DOI: 10.7546/nntdm.2025.31.4.747-760.

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