Geometry of cubic and quartic hypersurfaces over finite fields

Let Y subset of P-n be a cubic hypersurface defined over GF(q). Here, we study the Finite Field Nullstellensatz of order [q/3] for the set Y(q) of its GF(q)-points, the existence of linear subspaces of PG(n, q) contained in Y(q) and the possibility to join any two points of Y(q) by the union of two lines of PG(n, q) entirely contained in Y(q). We also study the existence of linear subspaces defined over GF(q) for the intersection of Y with s quadrics and for quartic hypersurfaces. (C) 2002 Elsevier Science (USA).