The number of rational points of a family of hypersurfaces over finite fields

Let Fq denote the finite field of odd characteristic p with q elements (q=pe, e∈N) and Fq⁎ represent the nonzero elements of Fq. In this paper, by using the Smith normal form of the index matrix, we give a formula for the number of rational points of the following family of hypersurface over Fq:∑j=0t−1∑i=1rj+1−rjarj+ix1erj+i,1…xnj+1erj+i,nj+1−b=0, where the integers t>0, r0=0