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

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 $\mathcal{H}^n$.

Theorems & Definitions (89)

• Definition 1.1
• Theorem 1.2: GD, MM, Smith
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3