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arXiv · 2403.05719

Generalized polynomials and hyperplane functions in $(\mathbb{Z}/p^k\mathbb{Z})^n$

Izabella Łaba·Charlotte Trainor
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Abstract

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 $\mathcal{H}^n$.

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Izabella Łaba, Charlotte Trainor. 2024-03-08. Generalized polynomials and hyperplane functions in $(\mathbb{Z}/p^k\mathbb{Z})^n$. https://arxiv.org/abs/2403.05719

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