正特征环上的Skolem-Mahler-Lech定理：更简短的证明与多维推广

中文摘要 AI 辅助

设$R$为交换环，$f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$是$R$-模$M$上的线性指数映射。Dong与Shafrir（2026）证明，当存在正整数$\ell$使得$\ell M = 0$时，$f$的零点集是可有效计算的$p$-正规集的交集，其中$p$取遍$\ell$的素因子；这推广了Derksen与Masser（2012）关于正特征域上$S$-单位方程解集的早期定理。本文有两个核心目的：其一，我们以Derksen-Masser定理为黑箱，给出Dong与Shafrir结果的更简短证明，该证明还能将零点集分解为域上线性指数方程零点集的仿射变换的正布尔组合；其二，我们证明了有限特征环上Skolem-Mahler-Lech定理的多维推广，具体而言，证明了满足$\ell M = 0$的$R$-模$M$上的任意$n$维线性递推序列的零点集，都是$\mathbb{N}^n$中可有效计算的$p$-正规集的交集，其中$p$取遍$\ell$的素因子；例如，这为判定两个经典线性递推序列在特征为$p^a$或$p^a q^b$（$p,q$为素数）的环上是否存在公共值提供了判定程序。

英文摘要

Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p$-normal sets, where $p$ ranges over the prime divisors of $\ell$. This generalizes an earlier theorem of Derksen and Masser (2012) on the solution set of $S$-unit equations over fields of positive characteristic. The purpose of this paper is twofold. First, we give a shorter proof of Dong and Shafrir's result, using the theorem of Derksen-Masser as a blackbox. Our proof also yields a decomposition of the zero set as a positive Boolean combination of affine transformations of zero sets of linear-exponential equations over fields. Second, we prove a multi-dimensional generalization of the Skolem-Mahler-Lech theorem over rings of finite characteristic. Specifically, we show that the zero set of every $n$-dimensional linear recurrence sequence over an $R$-module $M$ satisfying $\ell M = 0$ is the intersection of effectively computable $p$-normal sets (in $\mathbb{N}^n$), where $p$ ranges over the prime divisors of $\ell$. For example, this gives a decision procedure for whether two classical linear recurrence sequences have a common value over a ring of characteristic $p^a$ or $p^a q^b$, where $p$ and $q$ are primes.

发表机构

• Magdalen College, University of Oxford（牛津大学莫德林学院）
• Department of Mathematics, Ben-Gurion University of the Negev（内盖夫本-古里安大学数学系）

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