Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 223–247 | Cite as

Cylindrical Dyck paths and the Mazorchuk–Turowska equation

  • Jonas T. Hartwig
  • Daniele Rosso


We classify all solutions (pq) to the equation \(p(u)q(u)=p(u+\beta )q(u+\alpha )\) where p and q are complex polynomials in one indeterminate u, and \(\alpha \) and \(\beta \) are fixed but arbitrary complex numbers. This equation is a special case of a system of equations which ensures that certain algebras defined by generators and relations are non-trivial. We first give a necessary condition for the existence of non-trivial solutions to the equation. Then, under this condition, we use combinatorics of generalized Dyck paths to describe all solutions and a canonical way to factor each solution into a product of irreducible solutions.


Dyck path Generalized Weyl algebra 

Mathematics Subject Classification

05A10 16S35 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsUniversity of California RiversideRiversideUSA

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