Hilbert's Tenth Problem for some Noncommutative Rings

TL;DR

This paper investigates Hilbert's tenth problem in various noncommutative rings, establishing its undecidability in several cases and reducing it to known problems in commutative algebra.

Contribution

It extends the negative results of Hilbert's tenth problem to noncommutative rings like twisted polynomial rings and differential polynomial rings, and establishes reductions to commutative cases.

Findings

Hilbert's tenth problem is undecidable over certain noncommutative rings.

Recursively enumerable and Diophantine sets coincide in specific noncommutative rings.

Reductions to commutative cases enable transfer of undecidability results.

Abstract

We consider Hilbert's tenth problem for two families of noncommutative rings. Let $K$ be a field of characteristic $p$. We start by showing that Hilbert's tenth problem has a negative answer over the twisted polynomial ring $K\{\tau\}$ and its left division ring of fractions $K(\tau)$. We prove that the recursively enumerable sets and Diophantine sets of $\mathbb{F}_{q^n}\{\tau\}$ coincide. We reduce Hilbert's tenth problem over $\mathbb{F}_{q^n}\{\!\{\tau\}\!\}$ and $\mathbb{F}_{q^n}(\!(\tau)\!)$, the twisted version of the power series and Laurent series, to the commutative case. Finally, we show that the different models of $\mathbb{F}_q[T]$ in $K\{\tau\}$ we created are all equivalent in some sense which we will define. We then move on to the second family of rings, coming from differential polynomials. We show that Hilbert's tenth problem over $K[\partial]$ has a negative answer. We prove that Hilbert's tenth problem over the left division ring of fractions $K(\partial_1,\ldots,\partial_k)$ can be reduced to Hilbert's tenth problem over $C(t_1,\ldots,t_k)$ where $C$ is the field of constants of $K$. This gives a negative answer for $k \geq 2$ if the field of constants is $\mathbb{C}$ and for $k\geq 1$ if it is $\mathbb{R}$.