The Hilbert's-Tenth-Problem Operator

TL;DR

This paper studies the properties of the Hilbert's Tenth Problem operator over rings of integers with inverted primes, revealing its complex behavior in Turing degrees and its failure to preserve Turing equivalence.

Contribution

It demonstrates that the HTP operator generally does not preserve Turing degrees and shows the existence of sets where the operator's output differs significantly in computational complexity.

Findings

For almost all W, the jump W' is not diophantine in Z[W^{-1}].

HTP operator does not preserve Turing equivalence between different sets.

Reversals in Turing degrees of HTP outputs are possible, with some sets having strictly lower or higher degrees than others.

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 view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. For $W=\emptyset$, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump $\emptyset~\!'$ is Turing-equivalent to $HTP(\mathbb Z)$. More generally, $HTP(\mathbb Z[W^{-1}])$ is always Turing-reducible to $W'$, but not necessarily equivalent. We show here that the situation with $W=\emptyset$ is anomalous: for almost all $W$, the jump $W'$ is not diophantine in $\mathbb Z[W^{-1}]$. We also show that the $HTP$ operator does not preserve Turing equivalence: even for complementary sets $U$ and $\overline{U}$, $HTP(\mathbb Z[U^{-1}])$ and $HTP(\mathbb Z[\overline{U}^{-1}])$ can differ by a full jump. Strikingly, reversals are also possible, with $V<_T W$ but $HTP(\mathbb Z[W^{-1}]) <_T HTP(\mathbb Z[V^{-1}])$.