 |  |  | Commutativity of additive functions and rational functions |
Commutativity of additive functions and rational functions
For given fields K and bar (K) with a common subfield F
all the additive (or more general, all F-linear) functions
f:K -> bar (K) which fulfill a functional equation of the form
f((ax+b)/(cx+d))=(af(x)+b)/(gf(x)+d),
if both sides are defined, and
(
)ÎGL2(K) and
(
)ÎGL2(bar (K)), were determined by
F. Halter-Koch and L. Reich ([19]).
Under weak assumptions on the characteristic of F and
on the cardinality of F it can be proved that in the "generic case" f
must be a field monomorphism. This result can be seen as a generalization of
a well known theorem by Hua (cf. [8] Chapter 11, Theorem 1.3).
Later on the Möbius transformations were replaced by more general classes of
rational functions. (Cf. [20][21]).
It is an interesting task to investigate this functional equation (and the
expected characterization of field monomorphisms) in the situation K=bar (K)=R or K=bar (K)=C by applying analytic methods,
especially theorems similar to Ostrowski's theorem (on additive functions).
By doing this we will try to enlarge the class of rational functions as far
as possible.
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001
| | | Commutativity of additive functions and rational functions |