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Peer-reviewedCC BY-NC-ND 4.0Bulletin of the London Mathematical Society

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

  • Jade Nardi
  • Rodrigo San-José
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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.

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Cite

BibTeXRIS
BibTeX
@article{nardi2026maximum,
  title = {{Maximum number of zeroes of polynomials on weighted projective spaces over a finite field}},
  author = {Jade Nardi and Rodrigo San-José},
  journal = {Bulletin of the London Mathematical Society},
  year = {2026},
  month = jun,
  doi = {10.1112/blms.70417},
  eprint = {2507.22597},
  archivePrefix = {arXiv},
  url = {https://cnrs.hal.science/hal-05356702v2},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - JOUR
TI  - Maximum number of zeroes of polynomials on weighted projective spaces over a finite field
AU  - Jade Nardi
AU  - Rodrigo San-José
PY  - 2026
DA  - 2026/06/09
JO  - Bulletin of the London Mathematical Society
DO  - 10.1112/blms.70417
UR  - https://cnrs.hal.science/hal-05356702v2
AB  - 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.
SN  - 0024-6093
SN  - 1469-2120
ER  -