Algebraic curves P(x)-Q(y)=0 and functional equations
Complex Variables
2008-07-29 v2 Algebraic Geometry
Abstract
In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in C. For example, we show that for a generic pair of rational functions P,Q the first equation has no non-constant solutions f,g meromorphic on C whenever (\deg P-1)(\deg Q-1) \geq 2.
Cite
@article{arxiv.0804.0736,
title = {Algebraic curves P(x)-Q(y)=0 and functional equations},
author = {F. Pakovich},
journal= {arXiv preprint arXiv:0804.0736},
year = {2008}
}
Comments
final version accepted by Complex Var. and Elliptic Equ