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

TL;DR

This work develops a framework for solving S-unit equations over finitely presented modules M over the Laurent polynomial ring Z/TZ[X_1^{±},…,X_N^{±}], revealing that the solution set is effectively p-normal when T = p^e is a prime power and that, for general T, solvability reduces to linear-exponential Diophantine equations. The core methodology weaves coprimary decomposition, localization, pseudo Frobenius splitting, and finite-automaton analysis to translate arithmetic problems into combinatorial and semilinear descriptions, enabling effective decision procedures in many cases. A key contribution is the demonstration that S-unit equations over T-torsion modules are reducible to linear-exponential Diophantine systems, with decidability known when T has at most two distinct prime divisors (building on recent work by Karimov et al.). The framework not only advances understanding of S-unit equations in algebraic settings but also links to Submonoid Membership in wreath products and the Skolem problem, suggesting broad implications for computational number theory and group theory. Collectively, the results highlight a bridge between Diophantine problems and automata-theoretic representations, while also indicating that general decidability/undecidability for arbitrary T would require breakthroughs in number theory.

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.

Figures (10)

• Figure 1: An automaton for $\{2^k \mid k \in \mathbb{N}\}$.
• Figure 2: An automaton for $\{(a, 2a) \mid a \leq 0\}$.
• Figure 3: The automaton $\mathscr{U}$.
• Figure 4: A fragment of the "improved" automaton $\mathscr{U}$ for $\mathbb{Z}_{/4}[X^{\pm}]$.
• Figure 5: A transition of the automaton $\widetilde{\mathscr{U}}$.

Theorems & Definitions (111)

• Theorem 1.1: Derksen and Masser derksen2012linear
• Definition 1.2: reformulation of derksen2015linear
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1: wolper2000construction
• Theorem 3.1
• Corollary 3.1
• proof
• Example 3.2: Derksen's approach derksen2007skolem