  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 $\mathcal{H}^n$.
