General solution of functional equations defined by generic linear-fractional mappings F_1: C^N \to C^N and by generic maps birationally equivalent to F_1

TL;DR

This paper provides a comprehensive solution to a class of birational functional equations involving linear-fractional mappings in complex N-dimensional space, establishing connections with differential equations.

Contribution

It introduces a general solution framework for BFEs where the maps are either linear-fractional or birationally equivalent to a fixed map, extending previous results.

Findings

Explicit solutions for BFEs with degree one maps in any dimension.

Extension of solutions to BFEs with maps birationally equivalent to a base map.

Discussion of the relationship between BFEs and differential equations.

Abstract

We consider a system of birational functional equations (BFEs) (or finite-difference equations at w=m \in Z) for functions y(w) of the form: y(w+1)=F_n(y(w)), y(w):C \to C^N, n=deg(F_n(y)), F_n \in (\bf Bir}(C^N), where the map F_n is a given birational one of the group of all automorphisms of C^N \to C^N. The relation of the BFEs with ordinary differential equations is discussed. We present a general solution of the above BFEs for n=1,\forall N and of the ones with the map F_n birationally equivalent to F_1: F_n\equiv V\comp F_1\comp V^{-1}, \forall V \in (\bf Bir}(C^N).