There is no 290-Theorem for higher degree forms

TL;DR

This work extends the study of universality from quadratic forms to even-degree $m$-ic forms over totally real fields. It proves that universal $m$-ic forms do exist, but, unlike the quadratic case, there is no finite criterion—analogous to Bhargava–Hanke's 290-Theorem—that fully characterizes universality; rather, representation properties are governed by a flexible, unit-multiplicative structure and Waring-type bounds. Over $\mathbb{Z}$, the authors construct $Q_B$ to represent all $n\ge B+1$ while excluding $1,\dots,B$, and establish an equivalence between exclusion by a finite set and a multiplicative closure condition $ab^m\in\mathcal{A}\Rightarrow a\in\mathcal{A}$, with a rank bound of $<(B+1)(g(m)+1)$. Extending to totally real number fields, they adapt Waring-type results and geometry-of-numbers to show the existence of universal $m$-ic forms and, simultaneously, prove that finite criterion sets again cannot exist, via a construction that handles large-norm representations and finitely many small-norm classes. These findings illuminate the deeper complexity of higher-degree universality and the role of unit powers in representation theory over number fields.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 3.1: hilbert
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• proof