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

Author Aleksi Saarela

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LIPIcs.STACS.2026.77.pdf
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  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
Keywords
  • Combinatorics on words
  • word equation
  • parametric word

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Abstract

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.

Cite As Get BibTex

Aleksi Saarela. An Improved Version of Hmelevskii’s Theorem on Three-Variable Word Equations. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.77

Author Details

Aleksi Saarela
  • Department of Mathematics and Statistics, University of Turku, Finland

Funding

Supported by the Research Council of Finland under grant 339311.

References

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