Degree matrices and enumeration of rational points of some hypersurfaces over finite fields

Let f be a polynomial over the finite field Fq with degree matrix Df∈Z≥0n×m and N(f) be the number of Fq-rational points on the hypersurface defined by f=0. For an M∈Zn×m, let Df∼rqM denote that Df is row equivalent to M in the ring Z/(q−1)Z. Sun has originally found the formula for N(f) when n=m and 0