Polynomial equations for additive functions II. The mixed parameter case

Abstract

AbstractIn this sequence of work we investigate polynomial equations of additive functions. This is the continuation of the paper [5] entitled Polynomial equations for additive functions I. We consider here the solutions of the equation $$\begin{aligned} \sum _{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left( x\in \mathbb {F}\right) , \end{aligned}$$ ∑ 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 . 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.