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arXiv · 2404.06754

Mutual position of two smooth quadrics over finite fields

Shamil Asgarli·Chi Hoi Yip
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Abstract

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 Korchm\'{a}ros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in $\mathbb{P}^{n-1}$ with $n$ odd, where the answer is $q^{n-1}/4+O(q^{n-\frac{3}{2}})$.

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Shamil Asgarli, Chi Hoi Yip. 2024-04-10. Mutual position of two smooth quadrics over finite fields. https://doi.org/10.1007/s10623-025-01687-9

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