The Skolem Problem in rings of positive characteristic

TL;DR

This paper proves the decidability of the Skolem Problem for linear recurrence sequences over finitely generated commutative rings of positive characteristic, using recent advances in solving equations over torsion modules and multiplicative independence.

Contribution

It introduces an algorithm to determine zero terms in linear recurrence sequences over rings of positive characteristic, extending the decidability of the Skolem Problem to new algebraic structures.

Findings

Decidability of the Skolem Problem in rings of positive characteristic.

Zero set of sequences is a finite union of p-normal sets.

Algorithmic approach based on recent results in algebraic equations.

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).