Mutual position of two smooth quadrics over finite fields

TL;DR

The paper resolves a counting problem for the mutual position of two smooth quadrics over odd characteristic finite fields: the number of points external to one quadric and internal to the other is asymptotically a quarter of all points, $rac{q^{n-1}}{4}$, with error $O(q^{n-3/2})$ in odd dimension $n$. It achieves this by formulating internal/external points via dual quadrics and discriminants, then expressing the count as a quadratic character sum. Technical tools from character sum theory (Katz-type bounds) and geometry of quadrics (dual varieties, discriminants) are employed to isolate the main term and bound the error. The results extend the planar conic case to higher dimensions, offering precise asymptotics for all four relative-position sets and highlighting the role of discriminants in finite-geometry counts. The work has implications for understanding the distribution of polar-space types and provides explicit asymptotics for questions about mutual quadric incidences over finite fields.

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 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 $q^{n-1}/4+O(q^{n-\frac{3}{2}})$.

Theorems & Definitions (16)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5