An elementary bound for the number of points of a hypersurface over a finite field

We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve.Our bound is the best one for irreducible hypersurfaces that is linear on their degrees, because, for each finite field, there are at least two irreducible hypersurfaces of different degrees that reach our bound.