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

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.