HTP-complete rings of rational numbers

Abstract

For a ring R, Hilbert's Tenth Problem HT P (R) is the set of polynomial equations over R, in several variables, with solutions in R. We view HT P as an enumeration operator, mapping each set W of prime numbers to HT P (Z[W −1 ]), which is naturally viewed as a set of polynomials in Z[X 1 , X 2 , . . .]. It is known that for almost all W , the jump W ′ does not 1-reduce to HT P (R W ). In contrast, we show that every Turing degree contains a set W for which such a 1-reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for W ′ from HT P (Z[W −1 ]) can succeed uniformly on a set of measure 1, and regarding the consequences for the boundary sets of the HT P operator in case Z has an existential definition in Q.