Reduction from decidability over Q to decidability over Z for Hilbert’s Tenth Problem

Determine whether the existence of an algorithm that decides, for any multivariable polynomial with integer coefficients, whether it has a solution over the rationals Q necessarily implies the existence of an algorithm that decides whether such a polynomial has a solution over the integers Z. Equivalently, ascertain whether a positive solution to Hilbert’s Tenth Problem over Q would entail a positive solution to Hilbert’s Tenth Problem over Z.

Background

Hilbert’s Tenth Problem (H10) asks for an algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in a specified domain. It was shown by Davis–Putnam–Robinson–Matiyasevich that no such algorithm exists over Z. The status over Q remains a central question in definability and decidability in number theory.

The text notes a known direction of implication between the rational and integer versions and explicitly flags uncertainty about the reverse implication, which is directly tied to whether the Q-version would algorithmically resolve the Z-version. Establishing or refuting this implication is pivotal for understanding reductions between H10(Q) and H10(Z).