Monomial functions, normal polynomials and polynomial equations

TL;DR

This paper investigates the relationships between generalized monomial functions and polynomial equations, focusing on functional identities involving compositions with classical polynomials over fields.

Contribution

It introduces a framework for analyzing generalized monomial functions satisfying specific polynomial composition identities, extending classical results in polynomial functional equations.

Findings

Characterization of solutions to functional equations involving monomial functions and polynomials

Conditions under which generalized monomial functions satisfy polynomial composition identities

Extensions of classical polynomial functional equation results

Abstract

In this paper we consider generalized monomial functions $f, g\colon \mathbb{F}\to \mathbb{C}$ (of possibly different degree) that also fulfill \[ f(P(x))= Q(g(x)) \qquad \left(x\in \mathbb{F}\right), \] where $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ are given (classical) polynomials.