Cylindrical Dyck paths and the Mazorchuk-Turowska equation

TL;DR

This paper classifies all polynomial solutions to a functional equation related to algebraic structures, using combinatorics of generalized Dyck paths to describe and factor solutions.

Contribution

It provides a complete classification of solutions to the Mazorchuk-Turowska equation using combinatorial methods, revealing the structure of solutions and their irreducible factors.

Findings

All solutions characterized via generalized Dyck paths

A canonical factorization of solutions into irreducibles

Necessary conditions for existence of non-trivial solutions

Abstract

We classify all solutions (p,q) to the equation p(u)q(u)=p(u+b)q(u+a) where p and q are complex polynomials in one indeterminate u, and a and b 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.