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

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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 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.

Original languageEnglish
Pages (from-to)199-213
Number of pages15
JournalComplex Variables and Elliptic Equations
Volume56
Issue number1-4
DOIs
StatePublished - 1 Jan 2011

Keywords

  • Algebraic curves
  • Compositions
  • Functional equations
  • Meromorphic functions
  • Monodromy groups
  • Strong uniqueness polynomials

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