Further results on Hilbert's Tenth Problem

Hilbert's Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring DOUBLE-STRUCK CAPITAL Z of integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over DOUBLE-STRUCK CAPITAL Z. We show that there is no algorithm to determine for any P(z(1), horizontal ellipsis ,z(9)) is an element of DOUBLE-STRUCK CAPITAL Z[z(1), horizontal ellipsis ,z(g)] whether the equation P(z(1), horizontal ellipsis ,z(g)) = 0 has integral solutions with z(9) > 0. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z(1), horizontal ellipsis ,z(11)) = 0 (with integer coefficients) in 11 unknowns has integral solutions, which provides the best record on the original HTP over DOUBLE-STRUCK CAPITAL Z. We also show that there is no algorithm to test for any P(z(1), horizontal ellipsis ,z(17)) is an element of DOUBLE-STRUCK CAPITAL Z[z(1), horizontal ellipsis ,z(17)] whether P(z1(2), horizontal ellipsis , z1(2), horizontal ellipsis z2(17)) = 0 has integral solutions, and that there is a polynomial Q(z(1), horizontal ellipsis , z(20)) is an element of DOUBLE-STRUCK CAPITAL Z[z(1), horizontal ellipsis ,z(20)] such that {Q(z1(2), horizontal ellipsis , z20(2)): z(1), horizontal ellipsis , z(20) is an element of DOUBLE-STRUCK CAPITAL Z} boolean AND {0, 1, 2, horizontal ellipsis } coincides with the set of all primes.