Polynomial equations for additive functions I. The inner case

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 $$\sum _{i=1}^n{f}_i(x^{p_i})g_i(x^{q_i})=0(x\in \mathbb{F}),$$ where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_i,g_i\text{\hspace{0.17em}⁣}:\mathbb{F}\to \mathbb{C}$ are additive functions and $p_i,q_i$ are positive integers for all $i=1,\dots ,n$.