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

We consider the question of determining the maximum number of Fqrational 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. Yves Aubry Institut de Mathématiques de Toulon (IMATH), Université de Toulon, France and Aix Marseille Univ., CNRS, Centrale Marseille, I2M, Marseille, France. e-mail: [email protected] Wouter Castryck Laboratoire Painlevé, Université de Lille-1, Cité Scientifique, 59 655, Villeneuve d’Ascq, cedex, France and Departement Elektrotechniek imec-Cosic, KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium. e-mail: [email protected] Sudhir R. Ghorpade Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. e-mail: [email protected] Gilles Lachaud Aix Marseille Univ., CNRS, Centrale Marseille, I2M, Marseille, France. e-mail: [email protected] Michael E. O’Sullivan Department of Mathematics and Statistics, San Diego State University, San Diego, CA 921827720, USA. e-mail: [email protected] Samrith Ram Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India. e-mail: [email protected]