A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field

TL;DR

This paper establishes lower bounds on the rank of universal quadratic forms over certain totally real number fields, showing that for many fields, such forms cannot have arbitrarily low rank, especially as the field's discriminant increases.

Contribution

It extends previous work by providing new lower bounds for the rank of universal quadratic forms in monogenic, primitive, totally real fields, and demonstrates the finiteness of related interlacing polynomials.

Findings

Existence of lower bounds for the rank in specific number fields.

Infinitely many totally real cubic fields lack universal quadratic forms of a fixed rank.

Minimal rank of universal forms tends to infinity with the discriminant in certain quadratic fields.

Abstract

We show that if $K$ is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over $K$. In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number field grows. These results follow from the study of interlacing polynomials. Specifically, we show that there are only finitely many irreducible monic polynomials related to primitive number fields of a given degree, that have a bounded number of interlacing polynomials.