Effective Methods for Norm-Form Equations

Let $\alpha_1,\ldots,\alpha_k$ be linearly independent elements
of a number field $K$ of degree $n \ge k$, and let $m$ be an integer. The
equation $\mathrm{Norm}_{K/\mathbb{Q}} (x_1\alpha_1 + \cdots + x_k\alpha_k) = m$,
to be solved in integers, is called a `norm-form equation'. The case of
binary forms was solved by Thue in 1909, and the general case was resolved
by Schmidt in 1971 through his Subspace Theorem generalizing the work of
Thue-Siegel-Roth. Unfortunately these results are ineffective, and do not
provide any means of determining a bound on the height of exceptional
solutions-- in particular, they do not allow us to determine a complete
list of solutions for even a single norm-form equation.

Baker's theorem on linear forms in logarithms gave an effective version of
Thue's result for binary forms, and Vojta in his PhD thesis was able extend
this effectivity to three-variable norm-form equations under the assumption
that $K$ is totally complex and Galois. In this talk we discuss effective
resolution for certain norm-form equations in four and five variables,
extending the work of Vojta. In particular, we completely and effectively
resolve the question of norm-form equations over totally complex Galois
sextic fields. The results are motivated by joint work with Mike Bennett.