Permutations, hyperplanes and polynomials over finite fields

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 a 1 , . . . ,a n of GF(q), there are distinct field elements b 1 , . . . , b 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.

On types and classes of commuting matrices over finite fields

“…The first possibility is ruled out since S * can have no component of dimension 16. The only possible component of S * of dimension 24 is 1 (12,12) , and so, if our supposition is correct, then the primary types 2 (8,4) and 1 (12,12) must commute over F 2 . By Theorem 2.12, this is the case only if 1 (8,4) and 1 (6,6) commute.…”

“…Let the polynomials p, q, r, s, and t, the class C, and the type S be as in Example 2.14. Let D be the similarity class over F 2 with cycle type r (8,4) t (2,2,1) . The class type of D is T = 2 (8,4) 3 (2,2,1) .…”

“…Suppose that a separation T * of T commutes componentwise with a separation S * of S; then one of 2 (8) or 2 (8,4) is a component of T * . The first possibility is ruled out since S * can have no component of dimension 16.…”

Permutations over cyclic groups

“…Meanwhile in the group Z 3 2 , all multisets have a permutational sum which is zero. As it was briefly explained in [8], the problem has a connection to Snevily's conjecture [11], solved recently by Arsovski [3]. It would be natural to try to adapt the techniques which were successful for Snevily's problem, but our problems are apparently more difficult.…”

“…This theorem can be reformulated in the language of finite geometry and also have an application about the range of polynomials over finite fields. For more details, see [8].…”