On the number of points on a complete intersection over a finite field

Abstract Let V be a projective complete intersection defined over the finite field F q of q = p α elements and suppose it has dimension n and a singular locus of dimension d . We prove that the number of points of V with components in F q is equal to q n+1 −1 q−1 +O(q (n+d+1) 2 ) thus generalizing the well-known estimate of Deligne for the number of points on a non-singular complete intersection.