Universal quadratic forms and elements of small norm in real quadratic fields

TL;DR

This paper proves that for any positive integer M, infinitely many real quadratic fields lack M-ary universal quadratic forms, regardless of restrictions on cross coefficients.

Contribution

It establishes the non-existence of M-ary universal quadratic forms in infinitely many real quadratic fields for any M.

Findings

Infinitely many real quadratic fields do not admit M-ary universal quadratic forms.

The result holds without restrictions on the parity of cross coefficients.

Universal quadratic forms are not universally present in all real quadratic fields.

Abstract

For any positive integer M we show that there are infinitely many real quadratic fields that do not admit M-ary universal quadratic forms (without any restriction on the parity of their cross coefficients).