An optimal bound on the solution sets of one-variable word equations and its consequences

TL;DR

This paper establishes an optimal upper bound of three solutions for one-variable word equations with constants and provides a finite bound of 17 for certain three-variable systems with nonperiodic solutions, advancing understanding of word equation solution sets.

Contribution

It proves the conjectured bound of three solutions for one-variable word equations with constants and introduces the first finite bound for three-variable systems with nonperiodic solutions.

Findings

One-variable equations with constants have at most three solutions or infinitely many.

Three-variable systems without constants have at most 17 solutions if nonperiodic.

The bound of three solutions is proven to be optimal.

Abstract

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open.