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

TL;DR

This paper explores the possibility of a computable upper bound on the heights of rational solutions to certain Diophantine equations with finitely many solutions, proposing conjectures and their implications for decidability.

Contribution

It introduces conjectures on bounds for solutions' heights and shows these imply the existence of algorithms to bound solutions and decide finiteness of rational solutions.

Findings

Conjectures imply an algorithm to bound heights of solutions.

Conjectures imply decidability of finiteness of rational solutions.

Provides a link between height bounds and algorithmic decidability.

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.