S-Unit Equations in Modules and Linear-Exponential Diophantine Equations

Let T be a positive integer and M be a finitely presented module over the Laurent polynomial ring ℤ/T[X1±, …, XN±]. We consider S-unit equations over M: these are equations of the form x1 m1 + ⋯ + xK mK = m0, where the variables x1, …, xK range over the set of monomials (with coefficient 1) of ℤ/T[X1±, …, XN±]. When T is a power of a prime number p, we show that the solution set of an S-unit equation over M is effectively p-normal in the sense of Derksen and Masser (2015). This generalizes 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 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. S-unit equations in modules have direct connections to many problems in computational algebra such as finding sparse polynomials in ideals, identifying zeros of linear recurrence sequences, and deciding membership problems in metabelian groups. In particular, a direct consequence of our result is the decidability Submonoid Membership in wreath products of the form ℤ/pa qb ≀ ℤd.