Decidability of rational solvability for multivariate polynomial equations

Determine the decidability of the problem that, given a multivariate polynomial equation with coefficients in the rational numbers, decides whether the equation has a solution in the rational numbers. Establish whether a general algorithm exists to decide rational solvability for arbitrary multivariate polynomial equations over Q.

Background

The paper reduces loop synthesis with polynomial invariants to solving systems of polynomial equations whose solutions encode all admissible loop update maps. Since the implementation targets exact, finitely representable programs, the focus is on solutions with rational coefficients.

The authors note that, unlike the algebraic and real cases where general decision procedures exist (via Hilbert's Nullstellensatz and Tarski–Seidenberg, respectively), the existence of a general decision procedure over the rationals is a longstanding open problem in number theory. This limitation motivates their use of structural decomposition, numerical homotopy continuation, and SMT solvers as practical but incomplete strategies.