{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03127",
  "author_total": 2,
  "authors": [
    {
      "name": "Ruiwen Dong"
    },
    {
      "name": "Doron Shafrir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8n2sswp0h153y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03127",
  "fulltext_url": null,
  "id": "8n2sswp0h153y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03127",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7208753372",
  "pdf_url": "https://arxiv.org/pdf/2609.03127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8n2sswp0h153y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization",
  "title_text": "Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/8n2sswp0h153y",
  "venue": "arXiv",
  "versions": []
}