Algebraic curves P(x)-Q(y)=0 and functional equations

Details Algebraic curves P(x)-Q(y)=0 and functional equations Algebraic curves P(x)-Q(y)=0 and functional equations

2008-04-04

arxiv.org/abs/0804.0736v2

Mathematics

Complex Variables

final version accepted by Complex Var. and Elliptic Equ

Scientific paper

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.

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