Baire category theory and Hilbert's Tenth Problem inside $\mathbb{Q}$

TL;DR

This paper explores the computability of Hilbert's Tenth Problem over subrings of rationals using Baire category theory, revealing a deep connection between the complexity of HTP($bQ$) and the topological properties of these subrings.

Contribution

It introduces a novel application of Baire category theory to analyze the computability of HTP over subrings of rationals, establishing equivalences between computability and topological largeness.

Findings

HTP($\mathbb{Q}$) can compute any set C if and only if the class of subrings with this property is nonmeager.

Similar results are shown for 1-reducibility and Diophantine models of $\mathbb{Z}$.

The work links topological size (meagerness) of subring classes to their computational power.

Abstract

For a ring R, Hilbert's Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. We consider computability of this set for subrings R of the rationals. Applying Baire category theory to these subrings, which naturally form a topological space, relates their sets HTP(R) to the set HTP($\mathbb{Q}$), whose decidability remains an open question. The main result is that, for an arbitrary set C, HTP($\mathbb{Q}$) computes C if and only if the subrings R for which HTP(R) computes C form a nonmeager class. Similar results hold for 1-reducibility, for admitting a Diophantine model of $\mathbb{Z}$, and for existential definability of $\mathbb{Z}$.