A functional equation for monomial functions

TL;DR

This paper characterizes monomial functions over fields of characteristic zero satisfying a specific functional equation, revealing their structure in terms of homomorphisms and derivations.

Contribution

It determines the form of monomials satisfying a particular functional equation, extending classical results to this new context.

Findings

Monomials can be expressed using homomorphisms and higher-order derivations.

The functional equation constrains the structure of monomials significantly.

Results generalize classical additive function equations to monomials.

Abstract

Let $\mathbb{F}\subset \mathbb{K}$ be fields with characteristic zero, $n$ be a positive integer and $\kappa\in \mathbb{K}$. In this paper, we determine those monomials $f\colon \mathbb{F}\to \mathbb{K}$ of degree $n$ for which \[ f(x^{2})= \kappa\cdot x^{n}f(x) \] holds for all $x\in \mathbb{F}$. We show that similar to the classical results, where additive functions were considered, the monomial functions in the equation can be represented with the aid of homomorphisms and higher-order derivations.