线性递推序列的扭曲有理零点与局部-全局原理
Twisted Rational Zeros and Local-Global Principles for Linear Recurrence Sequences
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中文总结 AI 辅助
本文研究线性递推序列的Skolem猜想，通过扭曲有理零点与p-adic零点的关系，在p-adic Schanuel猜想下证明了两个互素LRS同时零点的局部-全局原理，并回答了相关开放问题。
中文摘要 AI 辅助
Skolem问题询问给定的线性递推序列（LRS）是否具有零项，它等价于证明Skolem-Mahler-Lech定理的有效版本，该定理指出非退化LRS具有有限多个零点。然而，Skolem问题的可判定性几十年来一直悬而未决。Bilu等人（2022）表明，在弱$p$-adic Schanuel猜想和Skolem猜想（也称为指数局部-全局原理）的前提下，简单LRS的Skolem问题是可判定的，其中后者断言一个LRS具有整数零点当且仅当它对每个整数$m$模$m$具有零点。本文致力于理解Skolem猜想。我们提供了一个例子，表明Skolem猜想的一个自然加强版本（将$m$限制为素数幂）是错误的。这一失败可通过LRS的$p$-adic零点（它们在Skolem-Mahler-Lech定理的证明中自然出现，并已被Bacik等人（2026）算法研究）与扭曲有理零点（由Bilu等人（2025）引入）之间的关系来解释。通过进一步研究这种关系，我们能够证明本文的主要结果：在$p$-adic Schanuel猜想的前提下，两个互素LRS的同时零点的局部-全局原理。证明该结果的一个基本步骤是刻画扭曲有理零点何时（或不是）无穷多个素数$p$的$p$-adic零点，我们无条件地做到了这一点。这也回答了Bilu等人（2025）的两个开放问题。最后，我们猜想扭曲有理零点的存在是上述加强的Skolem猜想可能失败的唯一方式，这得到了启发式论证的支持。
英文摘要
The Skolem Problem asks whether a given linear recurrence sequence (LRS) has a zero term, and is equivalent to proving an effective version of the Skolem-Mahler-Lech theorem, which states that a non-degenerate LRS has finitely many zeros. Decidability of the Skolem Problem however, has remained open for many decades. Bilu et al. (2022) showed that the Skolem Problem for simple LRS is decidable subject to the weak $p$-adic Schanuel Conjecture and the Skolem Conjecture (also known as the exponential local-global principle), the latter of which states that an LRS has an integer zero if and only if it has a zero modulo every integer $m$. This paper works towards understanding the Skolem Conjecture. We provide an example showing that a natural strengthening of the Skolem Conjecture, where we restrict $m$ to be a prime power, is false. This failure is accounted for by the relationship between $p$-adic zeros of LRS (which arise naturally in the proof of the Skolem--Mahler--Lech theorem and have been studied algorithmically by Bacik et al. (2026)) and twisted rational zeros (introduced by Bilu et al. (2025)). By studying this relationship further, we are able to prove the main result of this paper: a local-global principle for simultaneous zeros of two coprime LRS, subject to the $p$-adic Schanuel Conjecture. A fundamental step in proving this result is characterising when twisted rational zeros are (or are not) $p$-adic zeros for infinitely many primes $p$, which we do unconditionally. This also answers two open questions of Bilu et al. (2025). Finally, we conjecture that the existence of twisted rational zeros is the only way that the aforementioned strengthened Skolem Conjecture may fail, which is supported by a heuristic argument.
发表机构
- University of Oxford, UK（英国牛津大学）
- Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany（德国萨尔兰信息校园马克斯·普朗克软件研究所）
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