Quadratic equations in metabelian Baumslag-Solitar groups

Richard Mandel, Alexander Ushakov

Published 2023-02-14Version 1

For a finitely generated group $G$, the Diophantine problem over $G$ is the algorithmic problem of deciding whether a given equation $W(z_1,z_2,\ldots,z_k) = 1$ (perhaps restricted to a fixed subclass of equations) has a solution in $G$. We investigate the algorithmic complexity of the Diophantine problem for the class $\mathcal{C}$ of quadratic equations over the metabelian Baumslag-Solitar groups $BS(1,n)$. In particular, we prove that this problem is NP-complete whenever $n\neq 1$, and determine the algorithmic complexity for various subclasses (orientable, nonorientable etc.) of $\mathcal{C}$.