Northcott property and universality of higher degree forms

TL;DR

This paper investigates the finiteness and universality of higher degree forms over totally real fields, establishing that only finitely many extensions admit universal forms and none over certain infinite extensions with the Northcott property.

Contribution

It proves finiteness results for universal higher degree forms over totally real extensions and shows non-existence over infinite extensions with the Northcott property.

Findings

Finitely many totally real degree d extensions admit universal forms.

No universal forms exist over totally real infinite extensions with the Northcott property.

Provides conditions under which universality of higher degree forms fails or is limited.

Abstract

Let $K$ be a totally real number field, $d$ a positive integer, and $Q$ a higher degree form over $K$. We prove that there are at most finitely many totally real extensions $L/K$ of degree $d$ such that $Q$ over $L$ is universal. Further, we show that there are no universal forms over totally real infinite extensions of $\mathbb{Q}$ having the Northcott property.