Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: On Coset Coverings of Solutions of Homogeneous Cubic Equations over Finite Fields. | On Coset Coverings of Solutions of Homogeneous Cubic Equations over Finite Fields Ara Aleksanyan Department of Informatics and Applied Mathematics Yerevan State University Yerevan 375049 Armenia. alexara@ Mihran Papikian Department of Mathematics University of Michigan Ann Arbor MI 48109 . papikian@ Submitted April 21 1999 Accepted May 26 2001. MR Subject Classifications Primary 11T30 Secondary 05B40 Abstract Given a cubic equation XiyiZi x2y2z2 xnynzn b over a finite field it is necessary to determine the minimal number of systems of linear equations over the same field such that the union of their solutions exactly coincides with the set of solutions of the initial equation. The problem is solved for arbitrary size of the field. A covering with almost minimum complexity is constructed. 1 Introduction Throughout this paper Fq stands for a finite field with q elements and l- n for an n-dimensional linear space over Fq. If L is a linear subspace in l-q then the set ã L Ũ x I x 2 Lg ã 2 Fqn is a coset of the subspace L. An equivalent definition a subset N c l- n is a coset if whenever x1 x2 . xm are in N so is any affine combination m m of them . so is Xixi for any X1 . Xm in Fq such that Xi 1. It can be readily i 1 i 1 verified that any m-dimensional coset in l- n can be represented as a set of solutions of a certain system of linear equations over Fq of rank n m and vice versa. The purpose of this article is to estimate the minimum number of cosets of linear subspaces in Fq3n one must choose in order to precisely cover the set of all solutions of the homogeneous cubic equation x1 y1z1 x2y2z2 xnynzn b over Fq. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R22 1 The general covering problem was investigated by the hrst author in 1 - 3 in connection with linearized disjunctive normal forms of Boolean functions. A linearized disjunctive normal form . of a Boolean function f is a representation of the form f fl _ _ fp where each fj 2 n L n
