A class of functional equations for additive functions

The study of functional equations in which the unknown functions are assumed to be additive has a long history and continues to be an active area of research. Here we discuss methods for solving functional equations of the form (\(*\)) \(\sum _{j=1}^{k} x^{p_j}f_j(x^{q_j}) = 0\), where the \(p_j,q_j\) are non-negative integers, the \(f_j:R \rightarrow S\) are additive functions, S is a commutative ring, and R is a sub-ring of S. This area of research has ties to commutative algebra since homomorphisms and derivations satisfy equations of this type. Methods for solving all homogeneous equations of the form (\(*\)) can be found in Ebanks (Aequ Math 89(3):685-718, 2015), Ebanks (Results Math 73(3):120, 2018) and Gselmann et al. (Results Math 73(2):27, 2018). It seems that this fact may have been overlooked, judging by some results about a particular case of (\(*\)) in recent publications. We also present a new method for the homogeneous case by combining the results above with [6], and we show how to solve non-homogeneous equations of the form (\(*\)).