Polynomial equations for additive functions II. The mixed parameter case

Abstract In 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 aligned ₈=₁^nf₈ (x^p₈) g₈ (x) ^q₈= 0 (x F), aligned ∑ i = 1 n f i (x p i) g i (x) q i = 0 x ∈ F, where n is a positive integer, F C F ⊂ C is a field, f₈, g₈: F C f i, g i: F → C are additive functions and pᵢ, qᵢ p i, q i are positive integers for all i=1, , 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.