Annals of Computer Science and Information Systems, Volume 11

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

Krzysztof Molenda, Agnieszka Peszek, Maciej Sporysz, Apoloniusz Tyszka

Citation: Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 11, pages 249258 ()

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Abstract. The height of a rational number p/q is denoted by h(p/q) and equals 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 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 (if n=1), 2^(2^(n-2)) (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.

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