Polynomial equations for additive functions II. The mixed parameter case

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 ∑ ᵢ₌₁ⁿfᵢ(x^pᵢ)gᵢ(x)^qᵢ= 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, fᵢ, gᵢ: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.