The numbers of F-q-points of nonsingular hypersurfaces of a fixed degree in an odd -dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective 3 -space. (C) 2017 Elsevier Inc. All rights reserved.