Positive Moments Forever: Undecidable and Decidable Cases

Is there an algorithm to determine attributes such as positivity or non-zeroness of linear recurrence sequences? This long-standing question is known as Skolem's problem. In this paper, we study the complexity of an equivalent problem, namely the (generalized) moment membership problem for matrices. We show that this problem is decidable for orthogonal, unitary and real eigenvalue matrices, and undecidable for matrices over certain commutative and non-commutative polynomial rings. Our results imply that the positivity problem for simple unitary linear recurrence sequences is decidable, and is undecidable for linear recurrence sequences over the ring of commutative polynomials. As a byproduct, we prove a free version of Polya's theorem.