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：Skolem 问题要求判定给定的线性递推序列（LRS）是否存在零项。证明该问题的可判定性等价于给出 Skolem-Mahler-Lech 定理的一个有效证明，该定理断言非退化的 LRS 仅有有限多个零点。后一结果在九十多年前通过非构造性方法得以证明，其表明此类 LRS 仅存在有限多个 $p$-进零点。本文考虑判定给定 LRS 是否存在 $p$-进零点的问题，以及计算所有 $p$-进零点精确表示形式的相应函数问题。我们针对这两个问题提出了算法并报告了其实现情况。算法的输出是无条件正确的，其终止性依赖于 $p$-进 Schanuel 猜想（一个关于 $p$-进指数函数的标准数论假设）。虽然这些算法并未解决 Skolem 问题，但它们可在附加假设下用于寻找自然数和有理数零点。为说明这一点，我们应用所得结果证明了 Simultaneous Skolem 问题（判定两个互素的线性递推序列是否存在公共的自然数零点）的可判定性，该证明同样依赖于 $p$-进 Schanuel 猜想。