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

Abstract

In this article, 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 ∈ 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) ≥ 2.

Keywords:

• algebraic curves
• meromorphic functions
• functional equations
• compositions
• strong uniqueness polynomials
• monodromy groups

AMS Subject Classifications::

• 30D05
• 39B32

Acknowledgement

The author was supported by ISF, Grant No. 979/05.

Notes

Note

1. Prof. Yang kindly informed us that a similar result is obtained by a different method also in the forthcoming paper 17.