Positive Moments Forever: Undecidable and Decidable Cases

TL;DR

This paper explores the decidability of the generalized moment membership problem for matrices, establishing decidability for some classes and undecidability for others, with implications for positivity in linear recurrence sequences.

Contribution

It introduces new decidability results for matrix classes and extends Polya's theorem in a free setting, advancing understanding of Skolem's problem.

Findings

Decidability for orthogonal, unitary, and real eigenvalue matrices

Undecidability over certain polynomial rings

Positivity decidable for simple unitary linear recurrence sequences

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.