Polynomially linked additive functions—II

Abstract

We continue the study of additive functions $f_k:R\rightarrow F \;(1\le k\le n)$ linked by an equation of the form $\sum _{k=1}^n p_k(x)f_k(q_k(x))=0$, where the $p_k$ and $q_k$ are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case $\sum _{k=1}^n x^{m_k}f_k(x^{j_k})=0$ in which the $p_k$ and $q_k$ are monomials. In this case we show that if there is no duplication, i.e. if $(m_k,j_k)\ne (m_p,j_p)$ for $k\ne p$, then each $f_k$ is the sum of a linear function and a derivation of order at most $n-1$. Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way.