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

Abstract

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $E_{A,B}:A(x)-B(y)=0$, where $A,B\in \mathbb{C}(z)$. We also investigate "series" of curves $E_{A,B}$ 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 $B_i$, such that $\mathrm{deg}{B}_i\to \infty$ and all the curves $A(x)-B_i(y)=0$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $\mathbb{C}(z)/\mathbb{C}(A)$ has genus zero or one.

published version

References in corpus (1)

• Functional composition of polynomials: indecomposability, Diophantine equations and lacunary polynomials