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

作者：Pakovich F^*

来源：Complex Variables and Elliptic Equations, 2011, 56(1-4): 199-213.

DOI：10.1080/17476930903394838

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

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) = P(g), where c is an element of 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.

• 出版日期2011