# On solvability of certain equations of arbitrary length over torsion-free groups

## TL;DR

This paper proves the solvability of certain equations over torsion-free groups under specific coefficient relations and extends results to equations with higher powers of the variable, relating to the Kaplansky zero-divisor conjecture.

## Contribution

It establishes conditions for the solvability of arbitrary-length equations over torsion-free groups and generalizes to equations with higher powers of the variable.

## Key findings

- Solutions exist under a single coefficient relation.
- Results extend to equations with higher powers of the variable.
- Connections to the Kaplansky zero-divisor conjecture.

## Abstract

Let $G$ be a non-trivial torsion free group and $s(t)=g_{1}t^{\epsilon_{1}}g_{2}t^{\epsilon_{2}} \cdots g_{n}t^{\epsilon_{n}}=1 \; (g_{i} \in G,\ \epsilon_i=\pm 1)$ be an equation over $G$ containing no blocks of the form $t^{-1}g_{i}t^{-1}, \; g_{i} \in G$. In this paper we show that $s(t)=1$ has a solution over $G$ provided a single relation on coefficients of $s(t)$ holds. We also generalize our results to equations containing higher powers of $t$. The later equations are also related to Kaplansky zero-divisor conjecture \cite{K}.