Polynomial Equations for Additive Functions I: The Inner Parameter 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 \(\begin{aligned} \sum _{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left( x\in \mathbb {F}\right) , \end{aligned}\) $$\begin{array}{c}\hfill \sum _{i=1}^n{f}_i(x^{p_i})g_i(x^{q_i})=0\left(x,\in ,\mathbb{F}\right),\end{array}$$ where n is a positive integer, \(\mathbb {F}\subset \mathbb {C}\) $\mathbb{F}\subset \mathbb{C}$ is a field, \(f_{i}, g_{i}:\mathbb {F}\rightarrow \mathbb {C}\) $f_i,g_i:\mathbb{F}\to \mathbb{C}$ are additive functions and \(p_i, q_i\) $p_i,q_i$ are positive integers for all \(i=1, \ldots , n\) $i=1,\dots ,n$ .