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[ 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 U ¯ , HTP (ℤ[ U −1 ]) and H T P ( Z [ 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 ]).