  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 $HTP(\mathbb Z[W^{-1}]) <_T HTP(\mathbb
Z[V^{-1}])$.
