Polynomial equations for additive functions I. The inner case

TL;DR

This paper investigates polynomial equations involving additive functions over fields, leading to new characterization theorems for homomorphisms and derivations, focusing on equations of a specific polynomial form.

Contribution

It introduces new results on polynomial equations satisfied by additive functions, providing novel characterizations of homomorphisms and derivations in the inner case.

Findings

Derived new characterization theorems for homomorphisms.

Established conditions under which additive functions satisfy polynomial equations.

Extended understanding of additive functions in the context of polynomial equations.

Abstract

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered $$\sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left(x\in \mathbb{F}\right),$$ where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$.