On The Number of Points of A Hypersurface in Finite Projective Space

Abstract

In J.-P. Serre’s \(Lettre \;à\; M. Tsfasman\) \([3]\), 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.