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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.19518 |
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| _version_ | 1866914279627685888 |
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| author | Bleher, Frauke M. Chinburg, Ted Ding, Xuxi Heninger, Nadia Micciancio, Daniele |
| author_facet | Bleher, Frauke M. Chinburg, Ted Ding, Xuxi Heninger, Nadia Micciancio, Daniele |
| contents | Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $Δ(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another measure of the size of $K$ given by the covering radius $μ(K)$ of the ring of integers $O_K$ of $K$. Here $μ(K)$ is the $L^2$ radius $||V_2(K)||_2$ of the $L^2$ Voronoi cell $V_2(K)$ of $O_K$, where $V_2(K)$ is the set of points in $\mathbb{R} \otimes_{\mathbb{Q}} K$ that are at least as close to the origin as they are to any non-zero element of $O_K$. To put a limit on what lower bounds one can prove for $μ(K)$ in terms of $n(K)$, we study infinite families of $K$ of increasing degree for which $μ(K)$ can be bounded above by an explicit power of $n(K)$. We also study analogous questions when the $L^2$ norm is replaced by the $L^\infty$ norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the radii of Voronoi cells of rings of integers Bleher, Frauke M. Chinburg, Ted Ding, Xuxi Heninger, Nadia Micciancio, Daniele Number Theory Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $Δ(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another measure of the size of $K$ given by the covering radius $μ(K)$ of the ring of integers $O_K$ of $K$. Here $μ(K)$ is the $L^2$ radius $||V_2(K)||_2$ of the $L^2$ Voronoi cell $V_2(K)$ of $O_K$, where $V_2(K)$ is the set of points in $\mathbb{R} \otimes_{\mathbb{Q}} K$ that are at least as close to the origin as they are to any non-zero element of $O_K$. To put a limit on what lower bounds one can prove for $μ(K)$ in terms of $n(K)$, we study infinite families of $K$ of increasing degree for which $μ(K)$ can be bounded above by an explicit power of $n(K)$. We also study analogous questions when the $L^2$ norm is replaced by the $L^\infty$ norm. |
| title | On the radii of Voronoi cells of rings of integers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.19518 |