The Hilbert’s-Tenth-Problem Operator

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[X1, X2, . . .]. For W = ∅, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump ∅′ 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 = ∅ 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 U , HTP (Z[U−1]) and HTP (Z[U]) 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]).