Generalized polynomials and hyperplane functions in (Z/pᵏZ)ⁿ

For p prime, let Hⁿ be the linear span of characteristic functions of hyperplanes in (Z/pᵏZ)ⁿ. We establish new upper bounds on the dimension of Hⁿ over Z/pZ, or equivalently, on the rank of point-hyperplane incidence matrices in (Z/pᵏZ)ⁿ over Z/pZ. Our proof is based on a variant of the polynomial method using binomial coefficients in Z/pᵏZ as generalized polynomials. We also establish additional necessary conditions for a function on (Z/pᵏZ)ⁿ to be an element of Hⁿ.