Nonsingular points on hypersurfaces over $ \mathbb{F}_q $

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

“…As an application, the bound in Corollary 5 sharpens Theorem 2 in [7], concerning the existence of a smooth point on a hypersurface over F q .…”

“…Proof. In the proof of Theorem 2 in [7], replace the bound coming from the Cafure-Matera estimate (3) by the estimate from Corollary 5.…”

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

“…A classical problem is that of establishing conditions which imply that a given variety has an F q -rational point. Nevertheless, in several number-theoretical applications is not just an F q -rational point what is required, but a smooth F q -rational point (see, e.g., [26], [37], [38]).…”