The rank of new regular quadratic forms

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.