Diophantine sets of polynomials over number fields

TL;DR

This paper proves that in polynomial rings over certain number fields, recursively enumerable sets can be characterized as solutions to polynomial equations, extending the understanding of Diophantine sets in algebraic number theory.

Contribution

It demonstrates that recursively enumerable sets are diophantine within polynomial rings over recursive subrings of number fields, advancing the theory of Diophantine definability.

Findings

Recursively enumerable sets are diophantine in R[Z].

Extends Diophantine definability to polynomial rings over number fields.

Provides a new link between computability and algebraic structures.

Abstract

Let R be a recursive subring of a number field. We show that recursively enumerable sets are diophantine for the polynomial ring R[Z].