The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many $p$-adic zeros. In this paper we consider the problem of determining whether a given LRS has a $p$-adic zero, as well as the corresponding function problem of computing exact representations of all $p$-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the $p$-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the $p$-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the $p$-adic Schanuel Conjecture.

翻译：斯基奥莱姆问题旨在判定给定的线性递归序列（LRS）是否包含零项。该问题的可判定性等价于对斯基奥莱姆-马勒-莱赫定理给出有效的证明，该定理断言非退化线性递归序列只有有限个零点。后者定理于90多年前通过非构造性方法证明，表明此类线性递归序列仅有有限个$p$-进零点。本文考虑判定给定线性递归序列是否存在$p$-进零点的问题，以及相应的函数问题——计算所有$p$-进零点的精确表示。我们为这两个问题设计了算法并报告了实现结果。算法输出无条件正确，且终止性依赖于$p$-进沙努埃尔猜想（一种关于$p$-进指数函数的标准数论假设）。尽管这些算法未能解决斯基奥莱姆问题，但可在额外假设下用于寻找自然数零点和有理数零点。为说明这一点，我们应用本文结果证明了同步斯基奥莱姆问题（判定两个互质线性递归是否存在公共自然数零点）在$p$-进沙努埃尔猜想下的可判定性。