The rank of new regular quadratic forms

TL;DR

This paper proves that the rank of any new regular quadratic form is bounded by a universal constant, advancing understanding of the structure and limitations of such forms in number theory.

Contribution

It establishes a universal bound on the rank of new regular quadratic forms, a previously unknown fundamental property in the theory of quadratic forms.

Findings

The rank of any new regular quadratic form is bounded by an absolute constant.

Finiteness of regular ternary quadratic forms up to isometry is known.

Infinitely many equivalence classes of regular quadratic forms exist for rank ≥ 4.

Abstract

A (positive definite and integral) quadratic form $f$ is called regular if it represents all integers that are locally represented. It is known that there are only finitely many regular ternary quadratic forms up to isometry. However, there are infinitely many equivalence classes of regular quadratic forms of rank $n$ for any integer $n$ greater than or equal to $4$. A regular quadratic form $f$ is called new if there does not exist a proper subform $g$ of $f$ such that the set of integers that are represented by $g$ is equal to the set of integers that are represented by $f$. In this article, we prove that the rank of any new regular quadratic form is bounded by an absolute constant.