On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915292882403328 |
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| author | Singhal, Deepesh Lin, Yuxin |
| author_facet | Singhal, Deepesh Lin, Yuxin |
| contents | Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points in the vanishing set of $W$ as $W$ varies through all subspaces of $S_d(m)$ of dimension $r$. Beelen, Datta and Ghorpade had conjectured an exact formula of $e_r(d,m)$ when $q\geq d+1$. We prove that their conjectured formula is true when $q$ is sufficiently large in terms of $m,d,r$. The problem of determining $e_r(d,m)$ is equivalent to the problem of computing the $r^{th}$ generalized hamming weights of projective the Reed Muller code $PRM_q(d,m)$. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07702 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields Singhal, Deepesh Lin, Yuxin Algebraic Geometry Number Theory Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points in the vanishing set of $W$ as $W$ varies through all subspaces of $S_d(m)$ of dimension $r$. Beelen, Datta and Ghorpade had conjectured an exact formula of $e_r(d,m)$ when $q\geq d+1$. We prove that their conjectured formula is true when $q$ is sufficiently large in terms of $m,d,r$. The problem of determining $e_r(d,m)$ is equivalent to the problem of computing the $r^{th}$ generalized hamming weights of projective the Reed Muller code $PRM_q(d,m)$. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension $r$. |
| title | On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2311.07702 |