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

Yves Aubry (1, 2) , Wouter Castryck (3, 4) , Sudhir R Ghorpade (5) , Gilles Lachaud (1) , Michael E O 'Sullivan (6) , Samrith Ram (7)

1 I2M - Institut de Mathématiques de Marseille
2 IMATH - Institut de Mathématiques de Toulon - EA 2134
3 KU Leuven - Catholic University of Leuven = Katholieke Universiteit Leuven
4 LPP - Laboratoire Paul Painlevé - UMR 8524
5 IIT Bombay - Indian Institute of Technology Bombay
6 SDSU - San Diego State University
7 HRI - Harish-Chandra Research Institute

We consider the question of determining the maximum number of Fq-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field Fq, 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 Fq. 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.

Domaines: Géométrie algébrique [math.AG]; Théorie de l'information [cs.IT]

Yves Aubry, Wouter Castryck, Sudhir R Ghorpade, Gilles Lachaud, Michael E O 'Sullivan, et al.. Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory. Howe E., Lauter K., Walker J. Algebraic Geometry for Coding Theory and Cryptography, Association for Women in Mathematics Series, volume 9, Springer, Cham, pp.25-61, 2017, 978-3-319-63931-4. ⟨10.1007/978-3-319-63931-4_2⟩. ⟨hal-01478729v2⟩