Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization
Abstract
Let R be a commutative ring and f(a1, …, an) = ∑i=1k ri1a1 ⋯ rinan mi be a linear-exponential map over an R-module M. Dong and Shafrir (2026) showed that, when ℓ M = 0 for some ℓ ∈ ℕ>0, the zero set of f is the intersection of effectively computable p-normal sets, where p ranges over the prime divisors of ℓ. 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 ℓ M = 0 is the intersection of effectively computable p-normal sets (in ℕn), where p ranges over the prime divisors of ℓ. For example, this gives a decision procedure for whether two classical linear recurrence sequences have a common value over a ring of characteristic pa or pa qb, where p and q are primes.
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BibTeX
@misc{dong2026skolemmahlerlech,
title = {{Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization}},
author = {Ruiwen Dong and Doron Shafrir},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03127},
eprint = {2609.03127},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03127},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization
AU - Ruiwen Dong
AU - Doron Shafrir
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03127
UR - https://arxiv.org/abs/2609.03127
AB - Let R be a commutative ring and f(a_1, …, a_n) = ∑_i=1^k r_i1^a_1 ⋯ r_in^a_n m_i be a linear-exponential map over an R-module M. Dong and Shafrir (2026) showed that, when ℓ M = 0 for some ℓ ∈ ℕ_>0, the zero set of f is the intersection of effectively computable p-normal sets, where p ranges over the prime divisors of ℓ. 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 ℓ M = 0 is the intersection of effectively computable p-normal sets (in ℕ^n), where p ranges over the prime divisors of ℓ. 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.
ER -