Number fields without n-ary universal quadratic forms

TL;DR

This paper proves that for any fixed number of variables, there are infinitely many real quadratic fields where no universal quadratic form exists, highlighting limitations in representing numbers universally within these fields.

Contribution

It establishes the existence of infinitely many real quadratic fields lacking universal quadratic forms in any fixed number of variables.

Findings

Infinitely many real quadratic fields without M-variable universal quadratic forms.

Universal quadratic forms do not exist in these fields for any fixed M.

The result applies to all positive integers M.

Abstract

Given any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms in M variables.