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An application of random plane slicing to counting $\mathbb{F}_q$-points on hypersurfaces

Kaloyan Slavov

Abstract

Let $X$ be an absolutely irreducible hypersurface of degree $d$ in $\mathbb{A}^n$, defined over a finite field $\mathbb{F}_q$. The Lang-Weil bound gives an interval that contains $#X(\mathbb{F}_q)$. We exhibit explicit intervals, which do not contain $#X(\mathbb{F}_q)$, and which overlap with the Lang-Weil interval. In particular, we sharpen the best known lower and upper bounds for $#X(\mathbb{F}_q)$. The proof uses a combinatorial probabilistic technique.

An application of random plane slicing to counting $\mathbb{F}_q$-points on hypersurfaces

Abstract

Let be an absolutely irreducible hypersurface of degree in , defined over a finite field . The Lang-Weil bound gives an interval that contains . We exhibit explicit intervals, which do not contain , and which overlap with the Lang-Weil interval. In particular, we sharpen the best known lower and upper bounds for . The proof uses a combinatorial probabilistic technique.

Paper Structure

This paper contains 3 sections, 14 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Proof of the result for the "lower" forbidden interval
  3. Proof of the "upper" exclusion zone

Key Result

Theorem 1

Let $X$ be an absolutely irreducible plane curve of degree $d$. Then

Theorems & Definitions (23)

  • Theorem 1: Weil, Weil
  • Theorem 2: Ghorpade & Lachaud, GL
  • Theorem 3: Cafure & Matera, Cafure_Matera
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • proof
  • Corollary 7
  • proof
  • Theorem 8
  • ...and 13 more