[Paper Review] Intersection of two quadrics with no common hyperplane in $\mathbb{P}^{n}(\mathbb{F}_q)$}}

Frédéric A. B. Edoukou, San Ling+1 more|ArXiv.org|Jul 27, 2009

Finite Group Theory Research6 references3 citations

TL;DR

This paper proves a conjecture that the maximum number of $\mathbb{F}_q$-rational points in the intersection of two quadrics in $\mathbb{P}^n(\mathbb{F}_q)$ with no common hyperplane is $4q^{n-2} + \pi_{n-3}$, where $\pi_k = q^k + q^{k-1} + \cdots + 1$. The proof uses geometric decomposition and induction on dimension, building on prior bounds for low-dimensional cases and degenerate quadrics, and extends to a broader conjecture on algebraic sets of degree $d$ and dimension $s$: $|X(\mathbb{F}_q)| \leq dq^s + \pi_{s-1}$.

ABSTRACT

Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result states that $| \mathcal{Q}_1\cap \mathcal{Q}_2|\le 4q^{n-2}+π_{n-3}$. This result inspires us to establish the conjecture on the number of points of an algebraic set $X\subset {\mathbb{P}}^n(\mathbb{F}_q)$ of dimension $s$ and degree $d$: $|X(\mathbb{F}_q)|\le dq^s+π_{s-1}$.

Motivation & Objective

• To prove the conjecture that $|\mathcal{Q}_1 \cap \mathcal{Q}_2| \leq 4q^{n-2} + \pi_{n-3}$ for two quadrics in $\mathbb{P}^n(\mathbb{F}_q)$ with no common hyperplane.
• To establish the optimality of this bound by constructing explicit examples achieving equality.
• To generalize the bound to algebraic sets of dimension $s$ and degree $d$, proposing $|X(\mathbb{F}_q)| \leq dq^s + \pi_{s-1}$.

Proposed method

• Use of geometric decomposition: decompose the intersection into lower-dimensional components using linear subspaces $\mathbb{E}_t(\mathbb{F}_q)$ to reduce the problem to smaller dimensions.
• Inductive argument based on the order $w(\mathcal{Q}_1, \mathcal{Q}_2)$ of the pair of quadrics, defined as the minimal number of variables needed to express both.
• Leverage known bounds for non-degenerate and degenerate quadrics in low dimensions (e.g., $n=3,4$) as base cases.
• Apply results from the theory of resultants and Tsfasman-Serre-Sørensen bounds on hypersurfaces to control point counts.
• Use the structure of degenerate quadrics as cones over non-degenerate bases to analyze their intersections.
• Construct explicit examples achieving the bound $4q^{n-2} + \pi_{n-3}$ to confirm tightness.

Experimental results

Research questions

• RQ1 What is the best possible upper bound for the number of $\mathbb{F}_q$-rational points in the intersection of two quadrics in $\mathbb{P}^n(\mathbb{F}_q)$ with no common hyperplane?
• RQ2 Can the conjectured bound $4q^{n-2} + \pi_{n-3}$ be proven for all $n \geq 3$ and all finite fields $\mathbb{F}_q$?
• RQ3 Is the bound $|X(\mathbb{F}_q)| \leq dq^s + \pi_{s-1}$ optimal for any projective algebraic set $X$ of degree $d$ and dimension $s$?
• RQ4 How does the new bound compare to existing bounds such as those by Lachaud and Tsfasman-Serre-Sørensen?
• RQ5 Under what conditions does the intersection of two degenerate quadrics achieve the maximum possible number of rational points?

Key findings

• The paper proves that $|\mathcal{Q}_1 \cap \mathcal{Q}_2| \leq 4q^{n-2} + \pi_{n-3}$ for any two quadrics in $\mathbb{P}^n(\mathbb{F}_q)$ with no common hyperplane, and this bound is optimal.
• The bound is achieved when the quadrics are chosen such that their intersection contains a $\mathbb{P}^{n-3}$-component and a $4q^{n-2}$-component, as shown in explicit constructions.
• For $n=3$, the bound reduces to $4q + 1$, which matches the known optimal result for $\mathbb{P}^3(\mathbb{F}_q)$.
• For $n=4$, the bound is $4q^2 + q + 1$, which matches the previously conjectured and verified optimal bound.
• The authors confirm that the bound $4q^{n-2} + \pi_{n-3}$ is tighter than previous bounds by Leep and Schueller, especially in high dimensions.
• The generalized conjecture $|X(\mathbb{F}_q)| \leq dq^s + \pi_{s-1}$ is proposed as a refinement of Lachaud’s bound $d\pi_s$, and is shown to be superior in known cases.