One-equation reductions for the rational Hamilton quaternions and Lipschitz order

Definitions and mathematical statement

Shared notation.

$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.

Statement recorded in the supplied source.

Let $\mathbb H_{\mathbb Q}$ be the Hamilton quaternion algebra over $\mathbb Q$, and let $\mathcal O=\mathbb Z+i\mathbb Z+j\mathbb Z+k\mathbb Z$ be its Lipschitz order. Decide the solvability of arbitrary noncommutative polynomial equations with either rational or quaternion coefficients, and with unknowns constrained either to $\mathbb H_{\mathbb Q}$ or to $\mathcal O$. Also ask for existential definitions of $\mathbb Z$ in $\mathcal O$ and of $\mathbb Q$ in $\mathbb H_{\mathbb Q}$, and for the analogous decision questions in $M_n(\mathbb Z)$ and $M_n(\mathbb Q)$.

Formulation references: [S1], [S2].

Short English statement

Let $\mathbb H_{\mathbb Q}$ be the Hamilton quaternion algebra over $\mathbb Q$, and let $\mathcal O=\mathbb Z+i\mathbb Z+j\mathbb Z+k\mathbb Z$ be its Lipschitz order. Decide the solvability of arbitrary noncommutative polynomial equations with either rational or quaternion coefficients, and with unknowns constrained either to $\mathbb H_{\mathbb Q}$ or to $\mathcal O$. Also ask for existential definitions of $\mathbb Z$ in $\mathcal O$ and of $\mathbb Q$ in $\mathbb H_{\mathbb Q}$, and for the analogous decision questions in $M_n(\mathbb Z)$ and $M_n(\mathbb Q)$.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002344 (AIM-LOGIC-0120). Catalogue attribution under CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Original problem formulation and mathematical context. https://aimath.org/WWN/hilberts10th/hilberts10th.pdf.