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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2309.16978 |
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| _version_ | 1866909517412827136 |
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| author | Wang, Tianhao |
| author_facet | Wang, Tianhao |
| contents | An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field $\mathbb{F}_q$ with characteristic greater than $3$, Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is $\frac{1}{q}+O(q^{-2})$. We improve this result, showing that the density is exactly $\frac{q-1}{q^2-q+1}$.
We consider the problem of determining the density of pairs of conics satisfying the Poncelet $n$-gon condition for larger $n$. We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is $\frac{1}{q} + O(q^{-{3/2}})$. We show that when $n$ is an odd integer coprime to $q$, the density of pairs of smooth conics satisfying this condition is $\frac{d(n)-1}{q}+O(q^{-3/2})$, where $d(n)$ is the number of divisors of $n$. More generally, we conjecture that the density of pairs of conics satisfying the Poncelet $n$-gon condition is $d'(n)/q$ in general, where $d'(n)$ is the number of divisors of $n$ not equal to $1$ or $2$. Our argument involves analyzing the $n$-torsion points on a certain elliptic curve over the function field $K = \mathbb{F}_q(λ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16978 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition Wang, Tianhao Rings and Algebras An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field $\mathbb{F}_q$ with characteristic greater than $3$, Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is $\frac{1}{q}+O(q^{-2})$. We improve this result, showing that the density is exactly $\frac{q-1}{q^2-q+1}$. We consider the problem of determining the density of pairs of conics satisfying the Poncelet $n$-gon condition for larger $n$. We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is $\frac{1}{q} + O(q^{-{3/2}})$. We show that when $n$ is an odd integer coprime to $q$, the density of pairs of smooth conics satisfying this condition is $\frac{d(n)-1}{q}+O(q^{-3/2})$, where $d(n)$ is the number of divisors of $n$. More generally, we conjecture that the density of pairs of conics satisfying the Poncelet $n$-gon condition is $d'(n)/q$ in general, where $d'(n)$ is the number of divisors of $n$ not equal to $1$ or $2$. Our argument involves analyzing the $n$-torsion points on a certain elliptic curve over the function field $K = \mathbb{F}_q(λ)$. |
| title | Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2309.16978 |