Linear equations with monomial constraints and decision problems in abelian-by-cyclic groups

We show that it is undecidable whether a system of linear equations over the Laurent polynomial ring Z[X±] admit solutions where a specified subset of variables take value in the set of monomials ᶻ z ∈ Z\. In particular, we construct a finitely presented Z[X±]-module, where it is undecidable whether a linear equation Xz₁ f₁ + ⋯ + Xzₙ fₙ = f₀ has solutions z₁, …, zₙ ∈ Z. This contrasts the decidability of the case $n = 1$, which can be deduced from Noskov's Lemma. As applications, we show that there exists a finitely generated abelian-by-cyclic group in which the Knapsack Problem is undecidable, and in which the problem of solving quadratic equations is also undecidable. In contrast, we show that the problem of Coset Intersection is decidable in all finitely generated abelian-by-cyclic groups.