There is no 290-Theorem for higher degree forms

TL;DR

This paper investigates the universality of higher degree forms over totally real number fields, demonstrating their existence but showing they cannot be characterized by the 290-Theorem.

Contribution

It proves that universal forms of degree greater than 2 exist over these fields but are not describable by the 290-Theorem framework.

Findings

Universal forms of degree > 2 exist over totally real number fields.

Such forms cannot be characterized by the 290-Theorem.

The 290-Theorem does not extend to higher degree forms.

Abstract

We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290-Theorem of Bhargava-Hanke.