Intersection of two quadrics with no common hyperplane in ℙⁿ(\mathbbF_q)

2009/07/27 by Frédéric A. B. Edoukou, Edoukou, Frédéric A. B., San Ling +3
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.0907.4556

openalex publication_date 2009/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Q₁ and Q₂ be two arbitrary quadrics with no common hyperplane in ℙⁿ(\mathbbF_q). We give the best upper bound for the number of points in the intersection of these two quadrics. Our result states that | Q₁∩ Q₂|≤ 4q^n-2+π_n-3. This result inspires us to establish the conjecture on the number of points of an algebraic set X⊂ ℙⁿ(\mathbbF_q) of dimension s and degree d: |X(\mathbbF_q)|≤ dq^s+π_s-1.

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