Hilbert's 10th Problem for solutions in a subring of Q

TL;DR

This paper explores the decidability of Diophantine equations over subrings of rationals, linking the problem to the recursive enumerability of equations with finitely many solutions, and relates it to conjectures by Harvey Friedman.

Contribution

It establishes a connection between the solvability problem in subrings of Q and the recursive enumerability of equations with finitely many solutions, extending Matiyasevich's and Smoryński's theorems.

Findings

Positive solutions to H_{10}(R) imply recursive enumerability of equations with finitely many solutions in R.

For certain R, the problem H_{10}(R) is equivalent to Smoryński's theorem.

Conjectures by Friedman about rational solutions are shown to be equivalent in this framework.

Abstract

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smory\'nski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smory\'nski's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q.