Minimal universality criterion sets on the representations of quadratic forms

TL;DR

This paper investigates the properties of minimal universality criterion sets for quadratic forms, focusing on their uniqueness and structure, especially for binary quadratic forms and their subforms.

Contribution

It introduces the concept of minimal universality criterion sets for quadratic forms and proves their uniqueness in the case of most binary forms when considering subforms.

Findings

Minimal universality criterion sets are studied for quadratic forms.

For most binary quadratic forms, these sets are unique.

The paper characterizes properties of minimal sets in this context.

Abstract

For a set $S$ of (positive definite and integral) quadratic forms with bounded rank, a quadratic form $f$ is called $S$-universal if it represents all quadratic forms in $S$. A subset $S_0$ of $S$ is called an $S$-universality criterion set if any $S_0$-universal quadratic form is $S$-universal. We say $S_0$ is minimal if there does not exist a proper subset of $S_0$ that is an $S$-universality criterion set. In this article, we study various properties of minimal universality criterion sets. In particular, we show that for `most' binary quadratic forms $f$, minimal $S$-universality criterion sets are unique in the case when $S$ is the set of all subforms of the binary form $f$.