Representations of finite number of quadratic forms with same rank

Abstract

Let mn 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.

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Correspondence to Daejun Kim.

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This work was supported by the National Research Foundation of Korea (NRF-2019R1A2C1086347).

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Kim, D., Oh, BK. Representations of finite number of quadratic forms with same rank. Ramanujan J (2020). https://doi.org/10.1007/s11139-020-00314-6

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Keywords

  • Quadratic forms
  • Representations

Mathematics Subject Classification

  • 11E12
  • 11E20
  • 11E25