Representations of finite number of quadratic forms with same rank

2019/12/30 by Kim, Daejun, Oh, Byeong-Kweon
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.12930

Abstract

Let m, n be positive integers with m≤ n. Let κ(m,n) be the largest integer k such that for any (positive definite and integral) quadratic forms f₁,…,f_k of rank m, there exists a quadratic form of rank n that represents f_i for any i with 1≤ i ≤ k. In this article, we determine the number κ(m,n) for any integer m with 1≤ m≤ 8, except for the cases when (m,n)=(3,5) and (4,6). In the exceptional cases, it will be proved that 1≤ κ(3,5), κ(4,6)≤ 2. We also discuss some related topics.

Related

• The rank of new regular quadratic forms
• On the representations of a positive integer by certain classes of quadratic forms in eight variables
• A Study on the Number of Representations of an Integer into Certain Quadratic Forms
• Representations of integers by certain 2k-ary quadratic forms
• Representations by Quaternary Quadratic Forms with Coefficients 1,\n 2, 5 or 10