Characterization of field homomorphisms through Pexiderized functional equations

TL;DR

This paper characterizes field homomorphisms by analyzing solutions to Pexiderized functional equations involving additive functions, establishing that such solutions are linear combinations of field homomorphisms.

Contribution

It provides new characterization theorems for field homomorphisms based on solutions to specific functional equations involving additive functions.

Findings

Solutions are linear combinations of field homomorphisms.

Functional equations characterize field homomorphisms.

Results apply to arbitrary natural numbers n.

Abstract

The aim of this paper is to prove characterization theorems for field homomorphisms. More precisely, the main result investigates the following problem. Let $n\in \mathbb{N}$ be arbitrary, $\mathbb{K}$ a field and $f_{1}, \ldots, f_{n}\colon \mathbb{K}\to \mathbb{C}$ additive functions. Suppose further that equation \[ \sum_{i=1}^{n}f^{q_{i}}_{i}\left(x^{p_{i}}\right)=0 \qquad \left(x\in \mathbb{K}\right) \] is also satisfied. Then the functions $f_{1}, \ldots, f_{n}$ are linear combinations of field homomorphisms from $\mathbb{K}$ to $\mathbb{C}$.