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 X z | z ∈ Z. In particular, we construct a finitely presented Z[X ± ]-module, where it is undecidable whether a linear equation

This contrasts the decidability of the case n = 1, which can be deduced from Noskov's Lemma.

We apply this result to settle a number of problems in computational group theory. We show that it is undecidable whether a system of equations has solutions in the wreath product Z ≀ Z, providing a negative answer to an open problem of Kharlampovich, López and Miasnikov (2020). We show that there exists a finitely generated abelian-by-cyclic group in which the problem of solving a single (spherical) quadratic equation is undecidable, answering an open problem of Lysenok and Ushakov (2021). We also construct a finitely generated abelian-bycyclic group, different to that of Mishchenko and Treier (2017), in which the Knapsack Problem is undecidable. In contrast, we show that the problem of Coset Intersection is decidable in all finitely generated abelian-by-cyclic groups.