Characterization of field homomorphisms by functional equations, II

Summary.

Let K be a field containing $\Bbb Q$. We investigate pairs of additive functions $f,g: K \to K$ satisfying a functional equation¶¶$g \left( \frac{ax^n+\,b}{cx^n+\,d} \right) = \frac{af(x)^n+\,b} {cf(x)^n+\,d}\quad {\rm for}\quad n \in {\Bbb N}\quad {\rm and}\quad (^{a\,b}_{c\,d}) \in GL_2(K)$,¶where c=0 or (c ^-1 d)²=e ⁿ for some $e \in K$. If this functional equation holds for all $x \in K$ satisfying $cx^n + d \ne 0$ and $cf(x)^n +d \ne 0$, and if f is injective, then f(1)^-1 f is a field automorphism. On the way, we study functional equations of the form g(x ^ln) = W(x) f(x ^l)ⁿ for $l \in {\Bbb N},\,n \in {\Bbb Z} \setminus \{0,1\}$ and a function W taking only finitely many values, and functional equations of the form $[g(x^l)f (\frac 1{x^l}) ]^n = 1$.