Improved Lang--Weil bounds for a geometrically irreducible hypersurface over a finite field

Abstract

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we study the number of $\mathbb{F}_q$-points on the intersection of the given hypersurface with a random plane.