An Improved Version of Hmelevskii’s Theorem on Three-Variable Word Equations

Key Points

• The aim is to provide an enhanced understanding of solutions to constant-free three-variable word equations.
• Proving an improved version of Hmelevskii’s theorem.
• Demonstrating that three numerical parameters can define solutions.
• Establishing a framework that requires only two levels of nesting in the parametric solutions.
• Every constant-free three-variable word equation has a simpler parametric solution.
• Solutions can be expressed with just three numerical parameters and two nesting levels.
• The structure of the solution sets is considerably simplified compared to previous findings.