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 a"1,...,a"n of GF(q), there are distinct field elements b"1,...,b"n such that a"1b"1+...+a"nb"n=0. This implies the classification of hyperplanes lying in the union of the hyperplanes X"i=X"j 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.