S-unit equations in modules and linear-exponential Diophantine equations

Abstract

Let $T$ be a positive integer, and $\mathcal{M}$ be a finitely presented module over the Laurent polynomial ring $\mathbb{Z}_{/T}[X_1^{\pm}, \ldots, X_N^{\pm}]$. We consider S-unit equations over $\mathcal{M}$: these are equations of the form $x_1 m_1 + \cdots + x_K m_K = m_0$, where the variables $x_1, \ldots, x_K$ range over the set of monomials (with coefficient 1) of $\mathbb{Z}_{/T}[X_1^{\pm}, \ldots, X_N^{\pm}]$. When $T$ is a power of a prime number $p$, we show that the solution set of an S-unit equation over $\mathcal{M}$ is effectively $p$-normal in the sense of Derksen and Masser (2015), generalizing their result on S-unit equations in fields of prime characteristic. When $T$ is an arbitrary positive integer, we show that deciding whether an S-unit equation over $\mathcal{M}$ admits a solution is Turing equivalent to solving a system of linear-exponential Diophantine equations, whose base contains the prime divisors of $T$. Combined with a recent result of Karimov, Luca, Nieuwveld, Ouaknine and Worrell (2025), this yields decidability when $T$ has at most two distinct prime divisors. This also shows that proving either decidability or undecidability in the case of arbitrary $T$ would entail major breakthroughs in number theory. We mention some potential applications of our results, such as deciding Submonoid Membership in wreath products of the form $\mathbb{Z}_{/p^a q^b} \wr \mathbb{Z}^d$, as well as progressing towards solving the Skolem problem in rings whose additive group is torsion. More connections in these directions will be explored in follow up papers.