Maximum number of zeroes of polynomials on weighted projective spaces over a finite field

TL;DR

This paper determines the maximum number of rational points where a homogeneous polynomial can vanish on a weighted projective space over a finite field, confirming a conjecture under specific conditions and extending it.

Contribution

It proves a conjecture about the maximum number of zeros of polynomials on weighted projective spaces, extending previous bounds to more general degrees.

Findings

Confirmed a Serre-like bound for weighted projective spaces when the degree is divisible by the lcm of weights.

Extended the conjecture to include polynomials with degrees not restricted by divisibility.

Used footprint techniques, Delorme's reduction, and classical bounds in the proof.

Abstract

We compute the maximum number of rational points at which a homogeneous polynomial can vanish on a weighted projective space over a finite field, provided that the first weight is equal to one. This solves a conjecture by Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan and Ram, which stated that a Serre-like bound holds with equality for weighted projective spaces when the first weight is one, and when considering polynomials whose degree is divisible by the least common multiple of the weights. We refine this conjecture by lifting the restriction on the degree and we prove it using footprint techniques, Delorme's reduction and Serre's classical bound.