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, .,a(n) of GF(q), there are distinct field elements a(1), a(n), such that a(1)b(1) + +a(n)b(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. (C) 2010 Elsevier Inc. All rights reserve