Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

TL;DR

This paper proves that Hilbert's Tenth Problem is undecidable for certain holomorphy rings of function fields of characteristic zero, extending undecidability results to a broad class of algebraic structures.

Contribution

It establishes the undecidability of Hilbert's Tenth Problem for holomorphy rings of function fields in characteristic zero, generalizing previous results in number theory and algebraic geometry.

Findings

Hilbert's Tenth Problem is undecidable in these rings

Existence of specific elements in R with no algorithmic solution

Undecidability holds under recursive conditions on K

Abstract

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\Q(x_1,...,x_n)$ has solutions in $R$.