A note on pencils of norm-form equations

TL;DR

This paper completely solves a family of norm-form equations parametrized by t, using linear forms in logarithms and elementary methods, extending previous partial results.

Contribution

It provides a complete characterization of solutions to a specific family of norm-form equations, employing a novel combination of transcendence bounds and elementary techniques.

Findings

All solutions to the family of equations are explicitly determined.

The methods combine lower bounds for linear forms in logarithms with elementary arguments.

The results extend prior partial solutions to a full classification.

Abstract

We find all solutions to the parametrized family of norm-form equations $x^3-(t^3-1)y^3+3(t^3-1)xy+(t^3-1)^2 = \pm 1$ studied by Amoroso, Masser and Zannier. Our proof relies upon an appeal to lower bounds for linear forms in logarithms and various elementary arguments.