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

TL;DR

This paper introduces a probabilistic combinatorial method to refine bounds on the number of finite field points on hypersurfaces, improving the accuracy of existing estimates by explicitly excluding certain intervals.

Contribution

It presents explicit intervals that do not contain the exact point count, sharpening previous bounds for hypersurfaces over finite fields.

Findings

Derived explicit bounds that exclude certain intervals from containing $#X( ext{F}_q)$

Improved the lower and upper bounds for the number of points on hypersurfaces

Applied probabilistic combinatorial techniques to algebraic geometry problems

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.