Giorgos Kapetanakis

Denote by F q $\mathbb F_q$ double struck upper F Subscript q the finite field of order q and by F q n $\mathbb F_{q^n}$ double struck upper F Subscript q Sub Superscript n its extension of degree n . Some a ∈ F q n $a\in \mathbb F_{q^n}$ a element of double struck upper F Subscript q Sub Superscript n is called primitive if it generates the multiplicative group F q n ∗ $\mathbb F_{q^n}^*$ double struck upper F Subscript q Sub Superscript n Superscript asterisk , and it is called q n / q $q^n/q$ q Superscript n Baseline divided by q -normal if its F q $\mathbb F_q$ double struck upper F Subscript q -conjugates form an F q $\mathbb F_q$ double struck upper F Subscript q -basis of F q n $\mathbb F_{q^n}$ double struck upper F Subscript q Sub Superscript n if the latter is viewed as an F q $\mathbb F_q$ double struck upper F Subscript q -vector space. Furthermore, some a ∈ F q n $a\in \mathbb F_{q^n}$ a element of double struck upper F Subscript q Sub Superscript n is called q n / q $q^n/q$ q Superscript n Baseline divided by q -completely normal if it is q n / q d $q^n/q^d$ q Superscript n Baseline divided by q Superscript d -normal for all d ∣ n $d\mid n$ d vertical bar n . In this work, we prove a new construction of sets of completely normal elements and we establish, under conditions, the existence of elements that are simultaneously primitive and q n / q $q^n/q$ q Superscript n Baseline divided by q -completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen.

In 2022, S. D. Cohen and the two authors introduced and studied the concept of (r,n)-freeness on finite cyclic groups G for suitable integers r, n, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of G. Combining this machinery with some character sum techniques, they explored the existence of points (x 0 ,y 0 ) on affine curves y n =f(x) defined over a finite field ???? whose coordinates are generators of the multiplicative cyclic group ???? *. In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension ???? of a finite field ???? with Q elements is a cyclic ????[x]-module induced by the Frobenius automorphism α↦α Q, and any generator of this module is said to be a normal element over ????. We introduce and study the concept of (f,g)-freeness on this module structure for suitable polynomials f,g∈????[x]. As a main application of the machinery developed in this paper, we study the existence of ???? p n-rational points in the Artin–Schreier curve ???? f :y p -y=f(x) whose coordinates are normal over the prime field ???? p and establish concrete results.

Let $\Fm$ be finite fields of order $q^m$, where $m\geq 2$ and $q$, a prime power. Given $\F$-affine hyperplanes $A_1,\ldots, A_m$ of $\Fm$ in general position, we study the existence of primitive element $α$ of $\Fm$, such that $f(α)$ is also primitive, where $ax^2+bx+c\in \Fm[x]$ ($a\neq 0$ and $b^2\neq 4ac$) in $\Fm$ and the primitive pair $(α, f(α))$ avoids each $A_i$. We establish results for fields of higher order.

Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of $f_1(x,y)$ and $f_2(x,y)$. In this article, we discuss the existence of triples $(α, β, f(α, β))$ over $\mathbb{F}_q$, where $α, β$ are primitive and $f(α, β)$ is an $r$-primitive element of $\mathbb{F}_q$. In particular, this implies the existence of $\mathbb{F}_q$-primitive points on the surfaces of the form $z^r = f(x,y)$. As an example, we apply our results on the unit sphere over $\mathbb{F}_q$.

Let ???? Q be the cyclic group of order Q, n a divisor of Q and r a divisor of Q/n. We introduce the set of (r,n)-free elements of ???? Q and derive a lower bound for the number of elements θ∈???? q for which f(θ) is (r,n)-free and F(θ) is (R,N)-free, where f,F∈???? q [x]. As an application, we consider the existence of ???? q-primitive points on curves like y n =f(x) and find, in particular, all the odd prime powers q for which the elliptic curves y 2 =x 3 ±x contain an ???? q-primitive point.