On quadratic Waring's problem in totally real number fields
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866914999699505152 |
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| author | Krásenský, Jakub Yatsyna, Pavlo |
| author_facet | Krásenský, Jakub Yatsyna, Pavlo |
| contents | We improve the bound of the $g$-invariant of the ring of integers of a totally real number field, where the $g$-invariant $g(r)$ is the smallest number of squares of linear forms in $r$ variables that is required to represent all the quadratic forms of rank $r$ that are representable by the sum of squares. Specifically, we prove that the $g_{\mathcal{O}_K}(r)$ of the ring of integers $\mathcal{O}_K$ of a totally real number field $K$ is at most $g_{\mathbb{Z}}([K:\mathbb{Q}]r)$. Moreover, it can also be bounded by $g_{\mathcal{O}_F}([K:F]r+1)$ for any subfield $F$ of $K$. This yields a sub-exponential upper bound for $g(r)$ of each ring of integers (even if the class number is not $1$). Further, we obtain a more general inequality for the lattice version $G(r)$ of the invariant and apply it to determine the value of $G(2)$ for all but one real quadratic field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_15243 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On quadratic Waring's problem in totally real number fields Krásenský, Jakub Yatsyna, Pavlo Number Theory 11E12, 11D85, 11E25, 11E39 We improve the bound of the $g$-invariant of the ring of integers of a totally real number field, where the $g$-invariant $g(r)$ is the smallest number of squares of linear forms in $r$ variables that is required to represent all the quadratic forms of rank $r$ that are representable by the sum of squares. Specifically, we prove that the $g_{\mathcal{O}_K}(r)$ of the ring of integers $\mathcal{O}_K$ of a totally real number field $K$ is at most $g_{\mathbb{Z}}([K:\mathbb{Q}]r)$. Moreover, it can also be bounded by $g_{\mathcal{O}_F}([K:F]r+1)$ for any subfield $F$ of $K$. This yields a sub-exponential upper bound for $g(r)$ of each ring of integers (even if the class number is not $1$). Further, we obtain a more general inequality for the lattice version $G(r)$ of the invariant and apply it to determine the value of $G(2)$ for all but one real quadratic field. |
| title | On quadratic Waring's problem in totally real number fields |
| topic | Number Theory 11E12, 11D85, 11E25, 11E39 |
| url | https://arxiv.org/abs/2112.15243 |