On the simplest sextic fields and related Thue equations

TL;DR

This paper completely solves a family of sextic Thue equations, showing that only trivial solutions exist for all integer parameters, thus advancing understanding of solutions to these algebraic equations.

Contribution

It provides a complete classification of solutions for a parametric family of sextic Thue equations, identifying that only trivial solutions occur.

Findings

Only trivial solutions exist for the family of equations.

The solutions are explicitly characterized.

The results apply for all integer parameters m.

Abstract

We consider the parametric family of sextic Thue equations \[ x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda \] where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.