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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.05719 |
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| _version_ | 1866914709015363584 |
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| author | Łaba, Izabella Trainor, Charlotte |
| author_facet | Łaba, Izabella Trainor, Charlotte |
| contents | For $p$ prime, let $\mathcal{H}^n$ be the linear span of characteristic functions of hyperplanes in $(\mathbb{Z}/p^k\mathbb{Z})^n$. We establish new upper bounds on the dimension of $\mathcal{H}^n$ over $\mathbb{Z}/p\mathbb{Z}$, or equivalently, on the rank of point-hyperplane incidence matrices in $(\mathbb{Z}/p^k\mathbb{Z})^n$ over $\mathbb{Z}/p\mathbb{Z}$. Our proof is based on a variant of the polynomial method using binomial coefficients in $\mathbb{Z}/p^k\mathbb{Z}$ as generalized polynomials. We also establish additional necessary conditions for a function on $(\mathbb{Z}/p^k\mathbb{Z})^n$ to be an element of $\mathcal{H}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized polynomials and hyperplane functions in $(\mathbb{Z}/p^k\mathbb{Z})^n$ Łaba, Izabella Trainor, Charlotte Combinatorics 05B20, 05B25, 05A10 For $p$ prime, let $\mathcal{H}^n$ be the linear span of characteristic functions of hyperplanes in $(\mathbb{Z}/p^k\mathbb{Z})^n$. We establish new upper bounds on the dimension of $\mathcal{H}^n$ over $\mathbb{Z}/p\mathbb{Z}$, or equivalently, on the rank of point-hyperplane incidence matrices in $(\mathbb{Z}/p^k\mathbb{Z})^n$ over $\mathbb{Z}/p\mathbb{Z}$. Our proof is based on a variant of the polynomial method using binomial coefficients in $\mathbb{Z}/p^k\mathbb{Z}$ as generalized polynomials. We also establish additional necessary conditions for a function on $(\mathbb{Z}/p^k\mathbb{Z})^n$ to be an element of $\mathcal{H}^n$. |
| title | Generalized polynomials and hyperplane functions in $(\mathbb{Z}/p^k\mathbb{Z})^n$ |
| topic | Combinatorics 05B20, 05B25, 05A10 |
| url | https://arxiv.org/abs/2403.05719 |