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An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

Authors Dirk Nowotka, Aleksi Saarela

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Dirk Nowotka
  • Department of Computer Science, Kiel University, 24098 Kiel, Germany
Aleksi Saarela
  • Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland

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Dirk Nowotka and Aleksi Saarela. An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 136:1-136:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • combinatorics on words
  • word equations
  • systems of equations


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