The Skolem Problem in rings of positive characteristic

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 $(\gamma_n)_{n \in \mathbb{N}} \in \mathcal{R}^{\mathbb{N}}$, determines whether $(\gamma_n)_{n \in \mathbb{N}}$ contains a zero term. Our proof is based on two recent results: Dong and Shafrir (2026) 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).