  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 $i=1, \ldots,
n$.
