On certain equations of arbitrary length over torsion-free groups

TL;DR

This paper investigates specific equations over torsion-free groups, demonstrating solutions exist under certain coefficient relations, which simplifies solving such equations of length eight or more.

Contribution

It provides new conditions under which complex equations over torsion-free groups have solutions, simplifying their analysis for lengths eight and above.

Findings

Solutions exist when two coefficient relations are satisfied.

Equations of length ≥8 can be solved more straightforwardly.

Results apply to all non-trivial torsion-free groups.

Abstract

Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold. Such equations appear for all lengths greater than or equal to eight and the results presented in this article can substantially simplify their solution.