Permutations, hyperplanes and polynomials over finite fields

Gács, András and Héger, Tamás and Nagy, Zoltán Lóránt and Pálvölgyi, Dömötör (2010) Permutations, hyperplanes and polynomials over finite fields. FINITE FIELDS AND THEIR APPLICATIONS, 16 (5). pp. 301-314. ISSN 1071-5797

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Abstract

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 reserved

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