Permutations, hyperplanes and polynomials over finite fields

Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a1,…,an of GF(q), there are distinct field elements b1,…,bn such that a1b1+⋯+anbn=0. This implies the classification of hyperplanes lying in the union of the hyperplanes Xi=Xj in a vector space over GF(q), and also the classification of those multisets for which all reduced polynomials of this range are of reduced degree q−2. The proof is based on the polynomial method.