$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed

TL;DR

This paper investigates the existence of $r$-primitive $k$-normal polynomials over finite fields with prescribed last two coefficients, providing conditions for existence and identifying exceptional cases for specific parameters.

Contribution

It establishes sufficient conditions for the existence of such polynomials and characterizes all exceptional pairs $(q,m)$ for 3-primitive 1-normal polynomials with $m \\geq 7$.

Findings

Proved a sufficient condition for the existence of $r$-primitive $k$-normal polynomials with prescribed coefficients.

Computed all exceptional pairs $(q,m)$ for 3-primitive 1-normal polynomials with $m \\geq 7$.

Extended understanding of the structure and existence criteria of these polynomials over finite fields.

Abstract

Let $\xi\in\mathbb{F}_{q^m}$ be an $r$-primitive $k$-normal element over $\mathbb{F}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $\xi$ is referred to be the $r$-primitive $k$-normal polynomial of $\xi$ over $\mathbb{F}_q$. In this article, we study the existence of an $r$-primitive $k$-normal polynomial over $\mathbb{F}_q$ such that the last two coefficients are prescribed. In this context, first, we prove a sufficient condition which guarantees the existence of such a polynomial. Further, we compute all possible exceptional pairs $(q,m)$ in case of $3$-primitive $1$-normal polynomials for $m\geq 7$.