Counting Problems for Orthogonal Sets and Sublattices in Function Fields
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| Format: | Preprint |
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2024
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| _version_ | 1866915811453566976 |
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| author | Aranov, Noy Soffer Behajaina, Angelot |
| author_facet | Aranov, Noy Soffer Behajaina, Angelot |
| contents | Let $\mathcal{K}=\mathbb{F}_q((x^{-1}))$. Analogous to orthogonality in the Euclidean space $\mathbb{R}^n$, there exists a well-studied notion of ultrametric orthogonality in $\mathcal{K}^n$. In this paper, we extend the work of Soffer-Aranov and Behajaina on counting problems related to orthogonality in $\mathcal{K}^n$. For example, we resolve an open question posed in Soffer-Aranov and Behajaina by bounding the size of the largest ``orthogonal sets'' in $\mathcal{K}^n$. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over $\mathcal{K}$. Finally, we also use ultrametric orthogonality to compute the number of sublattices of $\mathbb{F}_q[x]^n$ with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in $\mathcal{K}^n$. The resulting formulas depend crucially on successive minima. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting Problems for Orthogonal Sets and Sublattices in Function Fields Aranov, Noy Soffer Behajaina, Angelot Combinatorics Number Theory 11H99, 05D99 Let $\mathcal{K}=\mathbb{F}_q((x^{-1}))$. Analogous to orthogonality in the Euclidean space $\mathbb{R}^n$, there exists a well-studied notion of ultrametric orthogonality in $\mathcal{K}^n$. In this paper, we extend the work of Soffer-Aranov and Behajaina on counting problems related to orthogonality in $\mathcal{K}^n$. For example, we resolve an open question posed in Soffer-Aranov and Behajaina by bounding the size of the largest ``orthogonal sets'' in $\mathcal{K}^n$. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over $\mathcal{K}$. Finally, we also use ultrametric orthogonality to compute the number of sublattices of $\mathbb{F}_q[x]^n$ with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in $\mathcal{K}^n$. The resulting formulas depend crucially on successive minima. |
| title | Counting Problems for Orthogonal Sets and Sublattices in Function Fields |
| topic | Combinatorics Number Theory 11H99, 05D99 |
| url | https://arxiv.org/abs/2411.19406 |