Genus one factors of curves defined by separated variable polynomials

TL;DR

This paper establishes conditions under which algebraic curves defined by separated variable polynomials lack genus 0 or 1 components, and identifies when these conditions are both necessary and sufficient.

Contribution

It provides new sufficient and necessary conditions for the genus of algebraic curves defined by P(x)-Q(y)=0, extending previous results under specific hypotheses.

Findings

Conditions ensuring no genus 0 or 1 components

Necessary and sufficient conditions when deg(P)=deg(Q)

Application of Hypothesis I by Fujimoto

Abstract

We give some sufficient conditions on complex polynomials P and Q to assure that the algebraic plane curve P(x)-Q(y)=0 has no irreducible component of genus 0 or 1. Moreover, if deg (P)=deg (Q) and if both P, Q satisfy Hypothesis I introduced by H. Fujimoto, our sufficient conditions are necessary.