Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre.s inequality for the number of points of projective hypersurfaces over finite fields is included.