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

## TL;DR

This paper investigates the maximum number of rational points on hypersurfaces in weighted projective spaces over finite fields, proposes conjectures, provides partial results, and explores applications to coding theory.

## Contribution

It extends the classical problem of point counts to weighted projective spaces, introduces conjectures, and connects findings to coding theory applications.

## Key findings

- Proposed conjectures for point counts in weighted projective spaces
- Partial results supporting the conjectures
- Applications to coding theory

## Abstract

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.