The Skolem Problem in rings of positive characteristic
Ruiwen Dong, Doron Shafrir
TL;DR
This work establishes the decidability of the Skolem Problem for linear recurrence sequences over finitely generated commutative rings of positive characteristic. The authors decompose the problem via the prime-power factors of the characteristic and prove that the zero-set of any recurrence is effectively $p$-normal for each prime power using Dong–Shafrir’s result on $S$-unit equations in $p^e$-torsion modules. They then show that intersections of $p$-normal sets across distinct primes can be effectively expressed as finite unions of $p$-normal sets, leveraging recent results on Presburger arithmetic with power predicates. Consequently, the zero set of a recurrence over a ring of characteristic $T=p_1^{e_1}inom{ dots}{e_k}$ is effectively a finite union of $p_i$-normal sets, yielding an algorithm to decide whether a zero term occurs. The findings extend Derksen’s effective description from prime characteristic to arbitrary positive characteristic and provide a constructive framework for analyzing Skolem-type questions in rings beyond fields.
Abstract
We show that the Skolem Problem is decidable in finitely generated commutative rings of positive characteristic. More precisely, we show that there exists an algorithm which, given a finite presentation of a (unitary) commutative ring $\mathcal{R} = \mathbb{Z}_{/T}[X_1, \ldots, X_n]/I$ of characteristic $T > 0$, and a linear recurrence sequence $(γ_n)_{n \in \mathbb{N}} \in \mathcal{R}^{\mathbb{N}}$, determines whether $(γ_n)_{n \in \mathbb{N}}$ contains a zero term. Our proof is based on two recent results: Dong and Shafrir (2025) on the solution set of S-unit equations over $p^e$-torsion modules, and Karimov, Luca, Nieuwveld, Ouaknine, and Worrell (2025) on solving linear equations over powers of two multiplicatively independent numbers. Our result implies, moreover, that the zero set of a linear recurrence sequence over a ring of characteristic $T = p_1^{e_1} \cdots p_k^{e_k}$ is effectively a finite union of $p_i$-normal sets in the sense of Derksen (2007).