Maximum number of zeroes of polynomials on weighted projective spaces over a finite field
This solves a specific conjecture in finite field geometry for a restricted class of weighted projective spaces, providing an exact bound for zeroes of polynomials.
The authors compute the maximum number of rational points where a homogeneous polynomial can vanish on a weighted projective space over a finite field, solving a conjecture by Aubry et al. for the case where the first weight is one, and refining it by removing a restriction on the degree.
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.