Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 223–247

# Cylindrical Dyck paths and the Mazorchuk–Turowska equation

• Jonas T. Hartwig
• Daniele Rosso
Article

## Abstract

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.

## Keywords

Dyck path Generalized Weyl algebra

05A10 16S35

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