Characterization of field homomorphisms through Pexiderized functional equations

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 $\mathbb{C}$.