On the Solution Sets of Three-Variable Word Equations
: Saarela, Aleksi
Publisher: SPRINGER
: NEW YORK
: 2024
: Theory of Computing Systems
: THEORY OF COMPUTING SYSTEMS
: THEOR COMPUT SYST
: 68
: 6
: 1556
: 1571
: 16
: 1432-4350
: 1433-0490
DOI: https://doi.org/10.1007/s00224-024-10193-9
: https://doi.org/10.1007/s00224-024-10193-9
: https://research.utu.fi/converis/portal/detail/Publication/457699339
It is known that the set of solutions of any constant-free three-variable word equation can be represented using parametric words, and the number of numerical parameters and the level of nesting in these parametric words is at most logarithmic with respect to the length of the equation. We show that this result can be significantly improved in the case of unbalanced equations, that is, equations where at least one variable has a different number of occurrences on the left-hand side and on the right-hand side. More specifically, it is sufficient to have two numerical parameters and one level of nesting in this case. We also discuss the possibility of proving a similar result for balanced equations in the future.
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Open Access funding provided by University of Turku (including Turku University Central Hospital).
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