Linear equations with monomial constraints and decision problems in abelian-by-cyclic groups
arxiv(2024)
Abstract
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
X^z_1f_1 + ⋯ + X^z_nf_n =
f_0 has solutions z_1, …, z_n ∈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.
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