# Cylindrical Dyck paths and the Mazorchuk–Turowska equation

@article{Hartwig2015CylindricalDP, title={Cylindrical Dyck paths and the Mazorchuk–Turowska equation}, author={Jonas Torbj{\"o}rn Hartwig and Daniele Rosso}, journal={Journal of Algebraic Combinatorics}, year={2015}, volume={44}, pages={223-247} }

We classify all solutions (p, q) to the equation $$p(u)q(u)=p(u+\beta )q(u+\alpha )$$p(u)q(u)=p(u+β)q(u+α) where p and q are complex polynomials in one indeterminate u, and $$\alpha $$α and $$\beta $$β 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…

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