Positive Moments Forever: Undecidable and Decidable Cases
arXiv:2404.15053 · doi:10.1016/j.laa.2025.05.015
Abstract
We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and undecidability for matrices over certain commutative and non-commutative polynomial rings. As consequences, we deduce that positivity is decidable for simple unitary linear recurrence sequences and undecidable for linear recurrence sequences over commutative polynomial rings. As a byproduct, we also prove a free version of Polya's theorem.
17 pages, v2: close to published version