Here are two ways to define Hilbert's tenth problem over a ring $R$:

- Given a polynomial $p \in \mathbb Z[x_1,\ldots,x_n]$, can one decide whether it has a solution in $R^n$?
- Given a polynomial $p \in R[x_1,\ldots,x_n]$, can one decide whether it has a solution in $R^n$?

I am interested in the case where $R$ is the ring of integers of a number field. In Bjorn Poonen's survey, he explains in §10 that he's talking about (1) but all the papers he cites in Theorem 14.1, including his own, discuss (2) and as far as I can tell don't mention (1). So that would seem to imply that (1) and (2) are equivalent in this situation, but I can't find this stated anywhere.

**The question:** Is it true that (1) and (2) are equivalent when $R$ is the ring of integers of a number field? Is there a reference for this?

**Addendum:** A sufficient criterion for Hilbert's tenth problem to be undecidable over $R$ is that $\mathbb Z$ is Diophantine over $R$. That is, there is a polynomial $P(x,y_1,\ldots,y_n)$ such that for $x \in R$,
\begin{equation} \tag{$*$}\label{thing}
\exists y_1,\ldots,y_n \in R\text{ such that }P(x,y_1,\ldots,y_n)=0 \quad \iff \quad x \in \mathbb Z.
\end{equation}
In fact, every proof of the undecidability of Hilbert's 10th problem over a number ring $R$ uses this idea. So:

**Effectively equivalent question:** Given a polynomial $P \in R[x,y_1,\ldots,y_n]$ satisfying \eqref{thing}, where $R$ is the ring of integers of a number field, is it always possible to find a polynomial $P \in \mathbb Z[x,y_1,\ldots,y_n]$ (maybe for a different $n$) satisfying \eqref{thing}?

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