Descent methods for studying integer points on $y^{n}=f(x)g(x),\ n\ge 2$

TL;DR

This paper investigates the distribution and properties of integer solutions on certain superelliptic and hyperelliptic curves defined by polynomial equations of the form y^n=f(x)g(x), focusing on cases where the degree sum is at least 4.

Contribution

The work introduces new methods for analyzing integer points on these classes of algebraic curves, extending previous results to more general polynomial forms.

Findings

Characterization of integer points on specific superelliptic and hyperelliptic curves

Conditions under which integer solutions exist or are finite

New bounds or criteria for solutions based on polynomial degrees

Abstract

We study the integer points on superelliptic and hyperelliptic curves of the form $y^n=f(x)g(x),$ $n\ge 2, {\rm{deg}}{f}+{\rm{deg}}{g}\ge 4.$