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

Hmelevskii proved in 1971 that every constant-free three-variable word equation has a parametric solution. We prove an improved version of this result by showing that every such equation has a parametric solution using only three numerical parameters and with only two levels of nesting. This means that the structure of the solution sets of these equations is considerably simpler than has been known before.