On the number of points of a hypersurface in finite projective space (after J.-P. Serre)

In J.-P. Serre's Lettre et M. Tsfasman [3], an interesting bound for the maximal number of points on a hypersurface of the n-dimensional projective space PG(n, q) over the Galois field GF(q) with q elements is given. Using essentially the same combinatorial technique as in [3], we provide a bound which is relative to the maximal dimension of a subspace of PG(n, q) which is completely contained in the hypersurface. The lower that dimension, the better the bound. Next, by using a different argument, we derive a bound which is again relative to the maximal dimension of a subspace of PG(n, q) which is completely contained in the hypersurface. If that dimension increases for the latter case, the bound gets better. As such, the bounds are complementary