The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) $\langle u_n \rangle_{n=0}^\infty$ over the integers has a zero term, that is, whether there exists $n$ such that $u_n = 0$. Decidability of the problem is open in general, with the most notable positive result being a decision procedure for LRS of order at most 4. In this paper we consider a bounded version of the Skolem Problem, in which the input consists of an LRS $\langle u_n \rangle_{n=0}^\infty$ and a bound $N \in \mathbb N$ (with all integers written in binary), and the task is to determine whether there exists $n\in\{0,\ldots,N\}$ such that $u_n=0$. We give a randomised algorithm for this problem that, for all $d\in \mathbb N$, runs in polynomial time on the class of LRS of order at most $d$. As a corollary we show that the (unrestricted) Skolem Problem for LRS of order at most 4 lies in $\mathsf{coRP}$, improving the best previous upper bound of $\mathsf{NP}^{\mathsf{RP}}$. The running time of our algorithm is exponential in the order of the LRS -- a dependence that appears necessary in view of the $\mathsf{NP}$-hardness of the Bounded Skolem Problem. However, even for LRS of a fixed order, the problem involves detecting zeros within an exponentially large range. For this, our algorithm relies on results from $p$-adic analysis to isolate polynomially many candidate zeros and then test in randomised polynomial time whether each candidate is an actual zero by reduction to arithmetic-circuit identity testing.

翻译：斯科伦问题旨在判定给定的整数线性递推序列（LRS）$\langle u_n \rangle_{n=0}^\infty$ 是否包含零项，即是否存在 $n$ 使得 $u_n = 0$。该问题的可判定性在一般情况下尚未解决，最著名的正面结果是针对不超过4阶LRS的判定算法。本文研究斯科伦问题的有界版本，其输入包含一个LRS $\langle u_n \rangle_{n=0}^\infty$ 和一个边界值 $N \in \mathbb N$（所有整数均以二进制形式表示），任务是判定是否存在 $n\in\{0,\ldots,N\}$ 使得 $u_n=0$。我们提出一个随机化算法，该算法对所有 $d\in \mathbb N$ 在不超过 $d$ 阶的LRS类上均以多项式时间运行。作为推论，我们证明不超过4阶LRS的（无界）斯科伦问题属于 $\mathsf{coRP}$ 类，改进了先前已知的最佳上界 $\mathsf{NP}^{\mathsf{RP}}$。该算法的运行时间相对于LRS的阶数呈指数增长——鉴于有界斯科伦问题的 $\mathsf{NP}$ 困难性，这种依赖性似乎是必要的。然而，即使对于固定阶数的LRS，该问题仍涉及在指数级大范围内检测零点。为此，我们的算法利用 $p$ 进分析的结果来隔离多项式数量的候选零点，然后通过归约到算术电路恒等性测试，以随机化多项式时间检验每个候选点是否为零点。