On the number of elements with prescribed norm and trace

TL;DR

This paper investigates the number of elements in finite field extensions with prescribed norm and trace, improving bounds by connecting to Artin-Schreier curves and applying advanced algebraic geometry techniques.

Contribution

It introduces improved estimates for N_n(a,b) using Artin-Schreier curves and extends bounds for rational points on related varieties, with explicit calculations and applications.

Findings

Improved bounds for N_n(a,b) when n ≥ √q - 1.

Enhanced estimates for rational points on certain algebraic varieties.

Explicit calculations of element counts and applications to polynomial enumeration.

Abstract

Let F_q be the finite field with cardinality q, where q is a prime power. Given a finite field extension F_q^n over F_q and a,b in (F_q)^{*}, we investigate in this article the number N_n(a,b) of elements in F_q^n whose norm equals a and trace equals b. Our approach to probe N_n(a,b) is to connect it with the number of rational points on certain Artin-Schreier curve. After establish an improvement of the Hasse-Weil bound for that Artin-Schreier curve, we improve the known estimates for N_n(a,b) when (roughly speaking) n \geq \sqrt{q}-1. Moreover, we use this approach to improve the bound given by Moisio and Wan for the number of rational points on the toric Calabi-Yau variety studied by Rojas-Leon and Wan in 2011. We finish the paper with explicit calculations of N_n(a,b) and an application to the number of irreducible monic polynomials in an arithmetic progression.