Representations of finite number of quadratic forms with same rank

TL;DR

This paper determines the maximum number of quadratic forms of a fixed rank that can be simultaneously represented by a higher-rank quadratic form, for ranks up to 8, with some exceptions.

Contribution

It explicitly computes the value of (m,n) for all m8 8, except for two cases, advancing understanding of quadratic form representations.

Findings

(m,n) is determined for all m8 8, except (3,5) and (4,6).

In exceptional cases, (3,5) and (4,6) are between 1 and 2.

The paper discusses related topics in quadratic form representation theory.

Abstract

Let $m, n$ be positive integers with $m\le n$. Let $\kappa(m,n)$ be the largest integer $k$ such that for any (positive definite and integral) quadratic forms $f_1,\ldots,f_k$ of rank $m$, there exists a quadratic form of rank $n$ that represents $f_i$ for any $i$ with $1\le i \le k$. In this article, we determine the number $\kappa(m,n)$ for any integer $m$ with $1\le m\le 8$, except for the cases when $(m,n)=(3,5)$ and $(4,6)$. In the exceptional cases, it will be proved that $1\le \kappa(3,5), \ \kappa(4,6)\le 2$. We also discuss some related topics.