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

Abstract

In J.-P. Serre’s $LettreàM.Tsfasman$ $[3]$, 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.