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

## TL;DR

This paper establishes the tightest known upper bound on the number of rational points in the intersection of two quadrics in projective space over finite fields, and proposes a conjecture for algebraic sets of higher degree.

## Contribution

It provides the best upper bound for the intersection points of two quadrics with no common hyperplane and proposes a general conjecture for algebraic sets.

## Key findings

- Upper bound: |Q1 ∩ Q2| ≤ 4q^{n-2} + π_{n-3}
- Conjecture: |X(𝔽_q)| ≤ d q^s + π_{s-1} for algebraic sets
- Inspiration for future bounds on algebraic set points

## 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 $| \mathcal{Q}_1\cap \mathcal{Q}_2|\le 4q^{n-2}+\pi_{n-3}$. 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 $d$: $|X(\mathbb{F}_q)|\le dq^s+\pi_{s-1}$.