Mutual position of two smooth quadrics over finite fields
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909848606605312 |
|---|---|
| author | Asgarli, Shamil Yip, Chi Hoi |
| author_facet | Asgarli, Shamil Yip, Chi Hoi |
| contents | Given two irreducible conics $C$ and $D$ over a finite field $\mathbb{F}_q$ with $q$ odd, we show that there are $q^2/4+O(q^{3/2})$ points $P$ in $\mathbb{P}^2(\mathbb{F}_q)$ such that $P$ is external to $C$ and internal to $D$. This answers a question of Korchmáros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in $\mathbb{P}^{n-1}$ with $n$ odd, where the answer is $q^{n-1}/4+O(q^{n-\frac{3}{2}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mutual position of two smooth quadrics over finite fields Asgarli, Shamil Yip, Chi Hoi Algebraic Geometry Combinatorics Number Theory Primary: 51E15, 14G15, Secondary: 15A63, 14J70, 11T24 Given two irreducible conics $C$ and $D$ over a finite field $\mathbb{F}_q$ with $q$ odd, we show that there are $q^2/4+O(q^{3/2})$ points $P$ in $\mathbb{P}^2(\mathbb{F}_q)$ such that $P$ is external to $C$ and internal to $D$. This answers a question of Korchmáros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in $\mathbb{P}^{n-1}$ with $n$ odd, where the answer is $q^{n-1}/4+O(q^{n-\frac{3}{2}})$. |
| title | Mutual position of two smooth quadrics over finite fields |
| topic | Algebraic Geometry Combinatorics Number Theory Primary: 51E15, 14G15, Secondary: 15A63, 14J70, 11T24 |
| url | https://arxiv.org/abs/2404.06754 |