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Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization — arXiv:2609.03127

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.03127
License (as archived): CC0-1.0
Archived: 2026-10-17T03:02:09.980Z
SHA-256 of the archived text: 8c0cf9090bbd4aed2a2a6f0208591c1f3c138effe6ea7a916b63497e6ed8a5e9
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Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization — arXiv:2609.03127 — arXiv (en), https://arxiv.org/abs/2609.03127 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_d4f4e92a2a8a (Metraipolis archive dc_d4f4e92a2a8a, sha256 8c0cf9090bbd4aed2a2a6f0208591c1f3c138effe6ea7a916b63497e6ed8a5e9)

BibTeX

@misc{dcd4f4e92a2a8a,
  title = {Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization — arXiv:2609.03127},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.03127},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_d4f4e92a2a8a; sha256 8c0cf9090bbd4aed2a2a6f0208591c1f3c138effe6ea7a916b63497e6ed8a5e9},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.03127. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-04 in cs.LO).
Categorie arXiv: Number Theory (math.NT) ; Logic in Computer Science (cs.LO).
Pagina del lavoro: https://arxiv.org/abs/2609.03127 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T02:39:28.939Z dalla pagina https://arxiv.org/catchup/cs.LO/2026-09-04?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).