Quadratic equations in metabelian Baumslag–Solitar groups
Richard M. Mandel, Alexander Ushakov · International Journal of Algebra and Computation · 2023
For a finitely generated group G, the Diophantine problem over G is the algorithmic problem of deciding whether a given equation [Formula: see text] (perhaps restricted to a fixed subclass of equations) has a solution in G. In this paper, we investigate the algorithmic complexity of the Diophantine problem for the class [Formula: see text] of quadratic equations over the metabelian Baumslag–Solitar groups [Formula: see text]. We prove that this problem is [Formula: see text]-complete whenever [Formula: see text], and determine the algorithmic complexity for various subclasses (orientable, nonorientable, etc.) of [Formula: see text].