An application of random plane slicing to counting Fq-points on hypersurfaces

Let X be an absolutely irreducible hypersurface of degree d in An, defined over a finite field Fq. The Lang-Weil bound gives an interval that contains #X(Fq). We exhibit an explicit interval, which does not contain #X(Fq), and which overlaps with the Lang-Weil interval. In particular, we sharpen the best known nontrivial lower bound for #X(Fq). The proof uses a combinatorial probabilistic technique.