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 εC 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 (degP-1)(degQ-1)≥2.
- Algebraic curves
- Functional equations
- Meromorphic functions
- Monodromy groups
- Strong uniqueness polynomials