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

2025-07-30

AI Summary

This paper addresses the problem of determining the maximum number of rational points on the zero set of a homogeneous polynomial in a weighted projective space over a finite field. For the case where the first weight equals one, the exact maximum is fully determined. The work confirms, generalizes, and rigorously strengthens a conjecture by Aubry et al.: when the polynomial degree is divisible by the least common multiple of the weights, a Serre-type upper bound is attained. By combining the footprint bound, Delorme’s reduction, and Serre’s classical estimate, the divisibility condition on the degree is removed, yielding a unified, degree-independent upper bound formula. This constitutes the first precise and universally applicable bound on the number of zeros in weighted projective spaces over finite fields. Integrating tools from algebraic geometry and finite field theory, the result significantly advances the state of the art in point counting on algebraic varieties over finite fields.

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• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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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.

Problem

• Compute max zeroes of polynomials on weighted projective spaces
• Solve conjecture on Serre-like bound for specific weight conditions
• Refine and prove conjecture using footprint and Delorme techniques

Innovation

• Computes max rational points for vanishing polynomials
• Uses footprint techniques and Delorme's reduction
• Applies Serre's bound to weighted projective spaces