General solution of functional equations defined by generic linear-fractional mappings F1: CN CN and by generic maps birationally equivalent to F1
Abstract
We consider a system of birational functional equations (BFEs) (or finite-difference equations at w=m ∈ Z) for functions y(w) of the form: y(w+1)=Fn(y(w)), y(w):C CN, n=deg(Fn(y)), Fn ∈ ( Bir(CN), where the map Fn is a given birational one of the group of all automorphisms of CN CN. The relation of the BFEs with ordinary differential equations is discussed. We present a general solution of the above BFEs for n=1,∀ N and of the ones with the map Fn birationally equivalent to F1: Fn V F1 V-1, ∀ V ∈ ( Bir(CN).
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