Decidability of the Moment Positivity Problem over the Rationals

Determine whether the moment membership problem is decidable or undecidable when the ring is \mathcal{R} = \mathbb{Q} and the target set is \mathcal{P} = [0, \infty); equivalently, decide whether there exists an algorithm that, given a rational matrix A, determines whether \operatorname{tr}(A^n) \ge 0 for all n \in \mathbb{N}.

Background

The paper develops decidable and undecidable cases of the moment membership problem, which asks whether all moments \operatorname{tr}(A^n) lie in a prescribed notion of positivity.

Despite several positive results (e.g., for orthogonal/unitary matrices or special spectral conditions) and undecidability over certain polynomial rings, the decisional status over the rationals with standard nonnegativity remains unresolved and is highlighted as the central open question.