Characterization of field homomorphisms through Pexiderized functional equations

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 $\mathbb{C}$.