$(G,F)$-points on $\mathbb{Q}$-algebraic varieties
Yangcheng Li, Hongjian Li
TL;DR
The paper investigates rational $(G,F)$-points on $V(G)$, where $G\in\mathbb{Q}[x,y,z]$ and $F$ applies a univariate polynomial $f$ componentwise, by establishing when the set $\langle G,F\rangle$ is birational to explicit elliptic curves. For the cases $G=xy-z^{2}$ and $G=(x+y)z-2xy$ combined with several $f$-forms (including $f=ax^{2}+bx+c$, $f=ax+b+cx^{-1}$, and $f=x(ax^{2}+bx+c)$), the results provide concrete Weierstrass models $\mathcal{E}_1$–$\mathcal{E}_4$ and show that $\langle G,F\rangle$ is in bijection with these curves (except in degenerate genus-0 scenarios). Rank analysis demonstrates that the elliptic curves typically have positive rank, implying infinitely many rational $(G,F)$-points, while certain parameter relations yield rank-zero cases with explicit rational points. The work thus furnishes a constructive elliptic-curve framework to generate and study rational solutions to the associated Diophantine systems.
Abstract
Let $G\in \mathbb{Q}[x,y,z]$ be a polynomial, and let $V(G)$ be the $\mathbb{Q}$-algebraic variety corresponding to $G$, i.e., $V(G)=\{P\in\mathbb{Q}^3~|~G(P)=0\}$. 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 know 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 $\mathbb{Q}$-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 a $\mathbb{Q}$-algebraic variety. 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.