Polynomial equations for additive functions II

Abstract

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, $f_i,g_i:\mathbb{F}\to \mathbb{C}$ are additive functions and $p_i,q_i$ are positive integers for all $i=1,\dots ,n$. 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.

arXiv comment29 pages. arXiv admin note: text overlap with arXiv:2211.03605