On Katz's bound for number of elements with given trace and norm

TL;DR

This paper improves Katz's bound on the count of finite field elements with specified trace and norm by linking the problem to rational points on toric Calabi-Yau hypersurfaces and applying advanced cohomological methods.

Contribution

It introduces a novel approach by reducing the problem to cohomological calculations on toric Calabi-Yau hypersurfaces, leading to a tighter bound.

Findings

Enhanced bound on elements with given trace and norm

Application of cohomological techniques to finite field problems

Reduction of the problem to rational point counts on toric hypersurfaces

Abstract

In this note an improvement of the Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurface, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces.