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.

10 pages. This is a minor revision from the previous version

References in corpus (1)

• Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields

Cited by in corpus (1)

• The exceptional locus in the Bertini irreducibility theorem for a morphism