  In this sequence of work we investigate polynomial equations of additive
functions. We consider the solutions of equation \[
  \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0
  \qquad
  \left(x\in \mathbb{F}\right), \] 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
$i=1, \ldots, n$. Using the theory of decomposable functions we describe the
solutions as compositions of higher order derivations and field homomorphisms.
In many cases we also give a tight upper bound for the order of the involved
derivations. Moreover, we present the full description of the solutions in some
important special cases, too.
