Polynomial Equations for Additive Functions I: The Inner Parameter Case

Abstract

AbstractThe 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 $$\begin{aligned} \sum _{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left( x\in \mathbb {F}\right) , \end{aligned}$$ ∑ i = 1 n f i ( x p i ) g i ( x q i ) = 0 x ∈ F , where n is a positive integer, $$\mathbb {F}\subset \mathbb {C}$$ F ⊂ C is a field, $$f_{i}, g_{i}:\mathbb {F}\rightarrow \mathbb {C}$$ f i , g i : F → C are additive functions and $$p_i, q_i$$ p i , q i are positive integers for all $$i=1, \ldots , n$$ i = 1 , … , n .