We show that it is undecidable whether a system of linear equations over the Laurent polynomial ring $\mathbb{Z}[X^{\pm}]$ admit solutions where a specified subset of variables take value in the set of monomials $\{X^z \mid z \in \mathbb{Z}\}$. In particular, we construct a finitely presented $\mathbb{Z}[X^{\pm}]$-module, where it is undecidable whether a linear equation $X^{z_1} \boldsymbol{f}_1 + \cdots + X^{z_n} \boldsymbol{f}_n = \boldsymbol{f}_0$ has solutions $z_1, \ldots, z_n \in \mathbb{Z}$. 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 $\mathbb{Z} \wr \mathbb{Z}$, providing a negative answer to an open problem of Kharlampovich, L\'{o}pez and Miasnikov (2020). We show that there exists a finitely generated abelian-by-cyclic group in which the problem of solving a single quadratic equation is undecidable. We also construct a finitely generated abelian-by-cyclic 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.

翻译：我们证明了，判定劳伦多项式环 $\mathbb{Z}[X^{\pm}]$ 上的线性方程组是否在指定变量子集取值于单项式集合 $\{X^z \mid z \in \mathbb{Z}\}$ 的条件下有解是不可判定的。具体而言，我们构造了一个有限展示的 $\mathbb{Z}[X^{\pm}]$-模，其中判定线性方程 $X^{z_1} \boldsymbol{f}_1 + \cdots + X^{z_n} \boldsymbol{f}_n = \boldsymbol{f}_0$ 是否存在解 $z_1, \ldots, z_n \in \mathbb{Z}$ 是不可判定的。这与 $n = 1$ 情形的可判定性形成对比，后者可由 Noskov 引理推得。我们应用此结果解决了计算群论中的若干问题。我们证明了判定方程组在圈积 $\mathbb{Z} \wr \mathbb{Z}$ 中是否有解是不可判定的，从而对 Kharlampovich, López 和 Miasnikov (2020) 提出的一个公开问题给出了否定回答。我们证明存在一个有限生成的阿贝尔-循环群，其中求解单个二次方程的问题是不可判定的。我们还构造了一个不同于 Mishchenko 和 Treier (2017) 所给群的有限生成阿贝尔-循环群，其中背包问题是不可判定的。与之相对，我们证明了在所有有限生成的阿贝尔-循环群中，陪集交集问题都是可判定的。