On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields

Deepesh Singhal; Yuxin Lin

Paper

On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields

Abstract

Consider a finite field

and positive integers

with

. Let

be the

vector space of all homogeneous polynomials of degree

. Let

be the maximum number of

-rational points in the vanishing set of

varies through all subspaces of

of dimension

. Beelen, Datta and Ghorpade had conjectured an exact formula of

when

. We prove that their conjectured formula is true when

is sufficiently large in terms of

. The problem of determining

is equivalent to the problem of computing the

generalized hamming weights of projective the Reed Muller code

. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension