Quadrics as hyperplanes in finite affine geometries

AbstractIn PG(n, q), n even, the number of points on a nondegenerate quadric is (qn − 1)(q − 1), the same as the cardinality of the hyperplanes. In a previous article we showed that PG(n, q), n even, q odd, possesses a family of nondegenerate quadrics that act as hyperplanes, in the sense that the intersections of the former have the same cardinalities and structure as those of the latter. This leads to a family of (q + 1)-caps which are the “lines” of a new incidence structure on the points of the original geometry.In the present article we describe the situation in AG(n, q), q odd. A family of “affine quadrics” is presented, whose members have qn−1 points, the same as the cardinality of the hyperplanes. A natural one-to-one correspondence arises between the two families. The intersections of these affine quadrics provide a family of q-sets which behave like lines and which are either q-caps or lines in the original geometry. To arrive at this construction, one starts with a symmetric matrix Q over GF(q), of size n + 1 and with minimal polynomial (x − λ)n+1. This leads to a ring of matrices of form c0I + c1Q + … +cnQn, ci ϵ GF(q), and they define the quadrics in question