Minimal rank of primitively $n$-universal integral quadratic forms over local rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912161315422208 |
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| author | Oh, Byeong-Kweon Yoon, Jongheun |
| author_facet | Oh, Byeong-Kweon Yoon, Jongheun |
| contents | Let $F$ be a local field and let $R$ be its ring of integers. For a positive integer $n$, an integral quadratic form defined over $R$ is called primitively $n$-universal if it primitively represents all quadratic forms of rank $n$. It was proved in arXiv:2005.11268 that the minimal rank of primitively $1$-universal quadratic forms over the $p$-adic integer ring $\mathbb{Z}_p$ is $2$ if $p$ is odd, and $3$ otherwise. In this article, we completely determine the minimal rank of primitively $n$-universal quadratic forms over $R$ for any positive integer $n$ and any local ring $R$ such that $2$ is a unit or a prime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal rank of primitively $n$-universal integral quadratic forms over local rings Oh, Byeong-Kweon Yoon, Jongheun Number Theory 11E08, 11E20 Let $F$ be a local field and let $R$ be its ring of integers. For a positive integer $n$, an integral quadratic form defined over $R$ is called primitively $n$-universal if it primitively represents all quadratic forms of rank $n$. It was proved in arXiv:2005.11268 that the minimal rank of primitively $1$-universal quadratic forms over the $p$-adic integer ring $\mathbb{Z}_p$ is $2$ if $p$ is odd, and $3$ otherwise. In this article, we completely determine the minimal rank of primitively $n$-universal quadratic forms over $R$ for any positive integer $n$ and any local ring $R$ such that $2$ is a unit or a prime. |
| title | Minimal rank of primitively $n$-universal integral quadratic forms over local rings |
| topic | Number Theory 11E08, 11E20 |
| url | https://arxiv.org/abs/2412.14709 |