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

TL;DR

This paper uses geometric methods involving Riemann surface orbifolds to establish lower bounds on the genus of algebraic curves defined by A(x)-B(y)=0, and characterizes when infinite genus-zero series exist based on Galois closure properties.

Contribution

It introduces a geometric approach to bound the genus of algebraic curves of the form A(x)-B(y)=0 and characterizes the existence of infinite genus-zero series via Galois closure genus.

Findings

Lower bounds for the genus of A(x)-B(y)=0 curves are established.

Infinite genus-zero series exist iff the Galois closure of the extension has genus zero or one.

The approach links algebraic properties with geometric and topological invariants.

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 ${\rm deg}\, B_i \rightarrow \infty$ 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.