How can we show that any diophantine set is recursively enumerable?
To be r.e., we require that the characteristic function is recursive and thus we require set to be decidable.
Only thing hat may be helpful is the parametrization theorem, but not sure how.
A set $S \subset \mathbb{N}$ is Diophantine if there is an $k + 1$ polynomial $P$ such that $n \in S$ iff for some $k$-tuple $\vec{m}$, $P(n, \vec{m}) = 0$.
Suppose $S$ is Diophantine, generated by the $k+1$ polynomial $P$. Take $k+1$ tuples in turn (zig-zag through them so each is reached in a finite number of steps) and test to see whether of not they satisfy the given Diophantine condition: output the first element of the successful tuples. That's an algorithmic procedure. So, voilà, the Diophantine set can be effectively enumerated. So, by a labour-saving appeal to Church's Thesis, it is recursively enumerable. It is as unexciting as that.