Effective Methods for Norm-Form Equations

TL;DR

This paper extends effective methods for solving norm-form equations, especially over totally complex Galois sextic fields, building on prior work on Thue equations and Vojta's results.

Contribution

It introduces new effective techniques for norm-form equations in multiple variables, including a complete solution for totally complex Galois sextic fields.

Findings

Successfully resolved classes of norm-form equations over complex Galois sextic fields.

Extended Vojta's results to new cases of norm-form equations.

Provided effective methods for equations in more than two variables.

Abstract

While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.