## Abstract

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*(ℤ[*W*^{−1}]), which is naturally viewed as a set of polynomials in ℤ[*X*_{1}, *X*_{2},…]. For *W* = Ø, it is a famous result of Matijasevič, Davis, Putnam and Robinson that the jump Ø′ is Turing-equivalent to *HTP*(ℤ). More generally, *HTP*(ℤ[*W*^{−1}]) is always Turing-reducible to *W*′, but not necessarily equivalent. We show here that the situation with *W* = Ø is anomalous: for almost all *W*, the jump *W*′ is not diophantine in *HTP*(ℤ[*W*^{−1}]). We also show that the *HTP* operator does not preserve Turing equivalence: even for complementary sets *U* and \(\bar U\), *HTP*(ℤ[*U*^{−1}]) and \(HTP(\mathbb{Z}{[\bar U]^{ - 1}})\) can differ by a full jump. Strikingly, reversals are also possible, with *V* <_{T}*W* but *HTP*(ℤ[*W*^{−1}]) <_{T}*HTP*(ℤ[*V*^{−1}]).

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