On the height of solutions to norm form equations

TL;DR

This paper establishes bounds on the Weil height of solutions within each equivalence class for norm form equations over number fields, providing a quantitative measure of solution complexity.

Contribution

It proves that every nonempty equivalence class of solutions has a representative with Weil height bounded by a parameter-dependent expression.

Findings

Bound on Weil height for solutions in each equivalence class

Quantitative relationship between solution height and equation parameters

Extension of known results to broader norm form equations

Abstract

Let $k$ be a number field. We consider norm form equations associated to a full $O_k$-module contained in a finite extension field $l$. It is known that the set of solutions is naturally a union of disjoint equivalence classes of solutions. We prove that each nonempty equivalence class of solutions contains a representative with Weil height bounded by an expression that depends on parameters defining the norm form equation.