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.

翻译：是否存在一种算法来确定线性递推序列的正性或非零性等属性？这一长期悬而未决的问题被称为斯科莱姆问题。在本文中，我们研究了一个等价问题的复杂性，即矩阵的（广义）矩隶属问题。我们证明，该问题对于正交矩阵、酉矩阵和实特征值矩阵是可判定的，而对于某些交换和非交换多项式环上的矩阵是不可判定的。我们的结果表明，简单酉线性递推序列的正性问题是可判定的，而交换多项式环上的线性递推序列的正性问题则是不可判定的。作为副产品，我们证明了波利亚定理的一个自由版本。