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




Nowotka Dirk, Saarela Aleksi

PublisherSIAM PUBLICATIONS

2022

SIAM Journal on Computing

SIAM JOURNAL ON COMPUTING

SIAM J COMPUT

51

1

1

18

18

0097-5397

DOIhttps://doi.org/10.1137/20M1310448(external)

https://doi.org/10.1137/20M1310448(external)

https://research.utu.fi/converis/portal/detail/Publication/174890785(external)



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 has at most 17 equations. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open.

Last updated on 2024-26-11 at 20:19