On algebraic curves A(x) - B(y) = 0 of genus zero

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5 Scopus citations


Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form EA,B:A(x)-B(y)=0, where A, B∈ C(z). We also investigate “series” of curves EA , B of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions Bi, such that deg Bi→ ∞ and all the curves A(x) - Bi(y) = 0 are irreducible and have genus zero, exists if and only if the Galois closure of the field extension C(z) / C(A) has genus zero or one.

Original languageEnglish
Pages (from-to)299-310
Number of pages12
JournalMathematische Zeitschrift
Issue number1-2
StatePublished - 1 Feb 2018


  • Galois coverings
  • Rational points
  • Separated variable polynomials
  • Two-dimensional orbifolds

ASJC Scopus subject areas

  • Mathematics (all)


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