  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.
