Hilbert’s Tenth Problem over the rational numbers

Determine whether Hilbert’s Tenth Problem over the field of rational numbers Q is decidable; specifically, establish whether there exists an algorithm that, given any multivariate polynomial with integer coefficients, decides whether the equation has a solution in the rational numbers.

Background

Hilbert’s Tenth Problem asks for an algorithm to decide solvability of arbitrary Diophantine equations. While undecidability over the integers is known via the DPRM theorem, the status over the rational numbers remains unresolved. The paper highlights this long-standing gap explicitly, distinguishing it from the classical integer case.

The authors’ work focuses on formalizing constructions and universality over the integers, but the rational case is flagged as still open, underscoring a key frontier in the field.