The complexity of the equation solvability problem over semipattern groups

TL;DR

This paper introduces a polynomial-time algorithm for solving equations over certain semidirect product groups and nilpotent rings by representing groups as matrix groups and reducing the problem to field equations.

Contribution

It presents a novel polynomial-time algorithm for equation solvability over specific semidirect product groups and improves efficiency for nilpotent rings.

Findings

Polynomial-time algorithm for certain semidirect product groups

More efficient algorithm for nilpotent rings

Representation of groups as matrix groups to reduce problem complexity

Abstract

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.