Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?

Abstract

The height of a rational number $p/q$ is denoted by $h(p/q)$ and equals $\text{max}(|p|,|q|)$ provided p/q is written in lowest terms. The height of a rational tuple $(x_1,...,x_n)$ is denoted by $h(x_1,...,x_n)$ and equals $\text{max}(h(x_1),...,h(x_n))$. Let $G_n={x_i+1=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}$. Let $f(1)=1$, and let $f(n+1)=2^{(2^{(f(n))})}$ for every positive integer n. We conjecture: (1) if a system $S \subseteq G_n$ has only finitely many solutions in rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \leq {1 (\text{if} n=1), 2^{(2^{(n-2)})} (\text{if} n>1)}$; (2) if a system $S \subseteq G_n$ has only finitely many solutions in non-negative rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \leq f(2n)$. We prove: (1) both conjectures imply that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite; (2) both conjectures imply that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution.