[Paper Review] As Easy as $\mathbb Q$: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields

Kirsten Eisentraeger, Russell Miller+2 more|arXiv (Cornell University)|Jan 26, 2016

Computability, Logic, AI Algorithms11 references3 citations

TL;DR

This paper constructs computably presentable subrings $ R \subseteq \mathbb{Q} $ and subrings of number fields $ \mathcal{O}_{K,\mathscr{S}} $ such that Hilbert's Tenth Problem over $ R $, denoted $ \operatorname{HTP}(R) $, is Turing equivalent to $ \operatorname{HTP}(\mathbb{Q}) $. It achieves this by selecting sets of primes $ \mathscr{S} $ with specified lower densities (including 0 and any computable $ r \in [0,1] $), ensuring $ \operatorname{HTP}(R) \equiv_T \operatorname{HTP}(\mathbb{Q}) $, and extends these results to number fields, showing that $ \operatorname{HTP}(\mathcal{O}_{K,\mathscr{S}}) \equiv_T \operatorname{HTP}(K) \leq_T \operatorname{HTP}(\mathbb{Q}) $. The work provides a rich hierarchy of rings with undecidability degrees matching that of $ \mathbb{Q} $.

ABSTRACT

Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$ having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is Turing equivalent to $HTP(\mathbb Q)$. We are able to put several additional constraints on the rings $R$ that we construct. Given any computable nonnegative real number $r \leq 1$ we construct such a ring $R = Z[\frac1p : p \in S]$ with $S$ a set of primes of lower density $r$. We also construct examples of rings $R$ for which deciding membership in $R$ is Turing equivalent to deciding $HTP(R)$ and also equivalent to deciding $HTP(\mathbb Q)$. Alternatively, we can make $HTP(R)$ have arbitrary computably enumerable degree above $HTP(\mathbb Q)$. Finally, we show that the same can be done for subrings of number fields and their prime ideals.

Motivation & Objective

• To investigate the computational complexity of Hilbert's Tenth Problem ($\mathrm{HTP}$) over subrings of $\mathbb{Q}$ and number fields, particularly in relation to the unresolved decidability of $\mathrm{HTP}(\mathbb{Q})$.
• To construct subrings $ R = \mathbb{Z}[\mathscr{S}^{-1}] \subseteq \mathbb{Q} $ such that $\mathrm{HTP}(R) \equiv_T \mathrm{HTP}(\mathbb{Q}) $, despite $\mathscr{S}$ being infinite and co-infinite.
• To extend these constructions to rings of integers localized at sets of primes in number fields, showing analogous Turing equivalence results.
• To demonstrate that the complexity of $\mathrm{HTP}(R) $ can be tuned via the density of the prime set $\mathscr{S}$, including achieving lower density 0 or any computable $ r \in [0,1] $.
• To explore the interplay between membership in $ R $, $\mathrm{HTP}(R)$, and $\mathrm{HTP}(\mathbb{Q})$, showing cases where they are all Turing equivalent.

Proposed method

• Constructing subrings $ R = \mathbb{Z}[\mathscr{S}^{-1}] \subseteq \mathbb{Q} $ by inverting a computably enumerable (c.e.) set of primes $ \mathscr{S} $, where $ \mathscr{S} $ is chosen to have a specified lower natural density.
• Using the theory of diophantine definitions and models, particularly leveraging results from Robinson and Poonen, to relate the undecidability of $\mathrm{HTP}(R)$ to that of $\mathbb{Q}$.
• Applying Turing reducibility to compare $\mathrm{HTP}(R) $ with $\mathrm{HTP}(\mathbb{Q}) $, showing $\mathrm{HTP}(R) \equiv_T \mathrm{HTP}(\mathbb{Q}) $ via oracle constructions and reductions.
• Extending the framework to number fields $ K $, defining rings $ \mathcal{O}_{K,\mathscr{S}} $ as the ring of $ \mathscr{S} $-integers, and analyzing $\mathrm{HTP}(\mathcal{O}_{K,\mathscr{S}}) $ using analogous density and reducibility arguments.
• Proving that for any computable real $ r \in [0,1] $, there exists a c.e. set $ \mathscr{S} $ of $ K $-primes with lower density $ r $ such that $ \mathrm{HTP}(\mathcal{O}_{K,\mathscr{S}}) \equiv_T \mathrm{HTP}(K) \leq_T \mathrm{HTP}(\mathbb{Q}) $.
• Demonstrating that the set of all $ K $-primes can be partitioned into finitely or infinitely many disjoint c.e. sets $ \mathscr{S}_i $, each of upper density 1, such that $ \mathrm{HTP}(\mathcal{O}_{K,\mathscr{S}_i}) \equiv_T \mathrm{HTP}(K) $.

Experimental results

Research questions

• RQ1 Can subrings $ R \subseteq \mathbb{Q} $ be constructed such that $ \mathrm{HTP}(R) \equiv_T \mathrm{HTP}(\mathbb{Q}) $, even when $ R $ is not $ \mathbb{Q} $ itself?
• RQ2 What is the relationship between the density of the set of inverted primes $ \mathscr{S} $ and the computational complexity of $ \mathrm{HTP}(\mathbb{Z}[\mathscr{S}^{-1}]) $?
• RQ3 Can the construction of such rings be extended to rings of integers in number fields, and do similar Turing equivalence results hold?
• RQ4 Is it possible to make the membership problem in $ R $, $ \mathrm{HTP}(R) $, and $ \mathrm{HTP}(\mathbb{Q}) $ all Turing equivalent?
• RQ5 Can the complexity of $ \mathrm{HTP}(R) $ be made arbitrarily high above $ \mathrm{HTP}(\mathbb{Q}) $, or can it be tuned to any c.e. degree?

Key findings

• For every computable real number $ r \in [0,1] $, there exists a computably enumerable set $ \mathscr{S} $ of rational primes with lower density $ r $, such that $ \mathrm{HTP}(\mathbb{Z}[\mathscr{S}^{-1}]) \equiv_T \mathrm{HTP}(\mathbb{Q}) $.
• There exist subrings $ R = \mathbb{Z}[\mathscr{S}^{-1}] \subseteq \mathbb{Q} $ with $ \mathscr{S} $ of lower density 0 such that $ \mathrm{HTP}(R) \equiv_T \mathrm{HTP}(\mathbb{Q}) $, and membership in $ R $ is also Turing equivalent to $ \mathrm{HTP}(R) $.
• For any computably enumerable set $ B \subset \mathbb{Z}_{>0} $ with $ \mathrm{HTP}(\mathbb{Q}) \leq_T B $, there exists a ring $ R = \mathbb{Z}[\mathscr{S}^{-1}] $ with $ \mathscr{S} $ of lower density 0 such that $ \mathrm{HTP}(R) \equiv_T B $, allowing $ \mathrm{HTP}(R) \equiv_T \mathrm{HTP}(\mathbb{Q}) $.
• The set of all $ K $-primes in a number field $ K $ can be partitioned into finitely many pairwise disjoint c.e. sets $ \mathscr{S}_1, \dots, \mathscr{S}_m $, each of upper density 1, such that $ \mathrm{HTP}(\mathcal{O}_{K,\mathscr{S}_i}) \equiv_T \mathrm{HTP}(K) \leq_T \mathrm{HTP}(\mathbb{Q}) $ for each $ i $.
• There exist infinitely many pairwise disjoint c.e. sets $ \mathscr{S}_1, \mathscr{S}_2, \dots $ of $ K $-primes, each of lower density 0, such that $ \bigcup_j \mathscr{S}_j $ is the full set of $ K $-primes and $ \mathrm{HTP}(\mathcal{O}_{K,\mathscr{S}_j}) \equiv_T \mathrm{HTP}(K) $ for all $ j $.
• For any number field $ K $, there exists a c.e. set $ \mathscr{W} $ of $ K $-primes of lower density 0 such that $ \mathrm{HTP}(\mathbb{Q}) \geq_T \mathrm{HTP}(K) \equiv_T \mathrm{HTP}(\mathcal{O}_{K,\mathscr{W}}) $.