Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory

We consider the question of determining the maximum number of \(\mathbb{F}_{q}\)-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field \(\mathbb{F}_{q}\), or in other words, the maximum number of zeros that a weighted homogeneous polynomial of a given degree can have in the corresponding weighted projective space over \(\mathbb{F}_{q}\). In the case of classical projective spaces, this question has been answered by J.-P. Serre. In the case of weighted projective spaces, we give some conjectures and partial results. Applications to coding theory are included and an appendix providing a brief compendium of results about weighted projective spaces is also included.