The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory. Currently, decidability is only known for LRS of order at most 4. On the other hand, the sole known complexity result is NP-hardness, due to Blondel and Portier. A fundamental result in this area is the celebrated Skolem-Mahler-Lech theorem, which asserts that the zero set of any LRS is the union of a finite set and finitely many arithmetic progressions. This paper focuses on a computational perspective of the Skolem-Mahler-Lech theorem: we show that the problem of counting the zeros of a given LRS is #P-hard, and in fact #P-complete for the instances generated in our reduction.

翻译：Skolem问题是指：给定一个整数线性递归序列(LRS)，判断该序列是否包含零项。其可判定性是理论计算机科学和自动机理论中长期未决的开放问题。目前，仅已知阶数不超过4的LRS的可判定性。另一方面，唯一已知的复杂度结果是Blondel和Portier给出的NP难度。该领域的基本结果是著名的Skolem-Mahler-Lech定理，该定理断言任何LRS的零点集是有限集与有限个算术级数的并集。本文聚焦于Skolem-Mahler-Lech定理的计算视角：我们证明给定LRS的零点计数问题是#P难的，并且对于我们的归约所生成的实例而言，实际上是#P完全的。