Quadratic equations in metabelian Baumslag-Solitar groups

TL;DR

This paper studies the complexity of solving quadratic equations in metabelian Baumslag-Solitar groups, establishing NP-completeness for most cases and analyzing subclasses.

Contribution

It proves NP-completeness of the Diophantine problem for quadratic equations over $ extbf{BS}(1,n)$ groups when $n eq \pm 1$, and explores subclasses' complexities.

Findings

NP-complete for $n eq \pm 1$

Complexity varies across subclasses

Provides a detailed complexity classification

Abstract

For a finitely generated group $G$, the \emph{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 $G$. In this paper, we investigate the algorithmic complexity of the Diophantine problem for the class $\mathcal{C}$ of quadratic equations over the metabelian Baumslag-Solitar groups $\mathbf{BS}(1,n)$. We prove that this problem is $\mathbf{NP}$-complete whenever $n\neq \pm 1$, and determine the algorithmic complexity for various subclasses (orientable, nonorientable etc.) of $\mathcal{C}$.