Frobenius nonclassical hypersurfaces

TL;DR

This paper investigates Frobenius nonclassical hypersurfaces over finite fields, establishing bounds on their degrees, characterizing extremal cases, and exploring their geometric properties and rational points.

Contribution

It provides sharp degree bounds, characterizations of maximal degree hypersurfaces, and shows their rational points form blocking sets, advancing understanding of their structure.

Findings

Sharp degree bounds for Frobenius nonclassical hypersurfaces

Characterization of hypersurfaces attaining maximal degrees

Frobenius nonclassical hypersurfaces have rational points forming blocking sets

Abstract

A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius nonclassical if the image of every geometric point under the $q$-th Frobenius endomorphism remains in the unique hyperplane tangent to the point. In this paper, we establish sharp lower and upper bounds for the degrees of such hypersurfaces, give characterizations for those achieving the maximal degrees, and prove in the surface case that they are Hermitian when their degrees attain the minimum. We also prove that the set of $\mathbb{F}_q$-rational points on a Frobenius nonclassical hypersurface form a blocking set with respect to lines, which indicates the existence of many $\mathbb{F}_q$-points.