Israel Journal of Mathematics

, Volume 230, Issue 2, pp 693–713 | Cite as

The Hilbert’s-Tenth-Problem Operator

  • Kenneth Kramer
  • Russell MillerEmail author


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 = 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 <TW but HTP(ℤ[W−1]) <THTP(ℤ[V−1]).


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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsQueens College—City University of New YorkQueensUSA
  2. 2.Graduate Center—City University of New YorkNew YorkUSA

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