Orientable quadratic equations in free metabelian groups

TL;DR

This paper proves that solving orientable quadratic equations in free metabelian groups is decidable and NP-complete, with polynomial-time solutions when the number of variables is fixed.

Contribution

It establishes the decidability and computational complexity of the Diophantine problem for these equations, including NP-completeness and fixed-variable polynomial-time solvability.

Findings

Decidable for orientable quadratic equations in free metabelian groups

NP-complete in general case

Polynomial-time when number of variables is bounded

Abstract

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.