Combinatorial Theory

For \(p\) prime, let \(\mathcal{H}^n\) be the linear span of indicator 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 \(\mathcal{H}^n\).

Mathematics Subject Classifications: 05B20, 05B25, 05A10

Keywords: Hyperplanes, generalized polynomials, binomial coefficients