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Himangshu Hazarika

Publications and source records attributed to Himangshu Hazarika.

7 recordsLinked to original sources
arXiv2024

Existence of Special Types Primitive Pairs in Finite Fields Avoiding Affine Hyperplanes

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.

math.NT↗
arXiv2024

On existence of primitive normal elements of rational form over finite fields of even characteristic

Let $q$ be an even prime power and $m\geq2$ an integer. By $\mathbb{F}_q$, we denote the finite field of order $q$ and by $\mathbb{F}_{q^m}$ its extension degree $m$. In this paper we investigate the existence of a primitive normal pair $(α, \, f(α))$, with $f(x)= \dfrac{ax^2+bx+c}{dx+e} \in \mathbb{F}_{q^m}(x)$, where the rank of the matrix $F= \begin{pmatrix}a \, &b\, & c\\ 0\, &d \, &e \end{pmatrix}$ $\in M_{2 \times 3}(\Fm) $ is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for $\begin{pmatrix} 1 \, &1 \, & 0\\ 0\, &1 \, &0 \end{pmatrix}$ if $q=2$ and $m$ is odd, and then we provide an explicit list of possible and genuine exceptional pairs $(q,m)$.

math.NT↗
arXiv2024

Primitive normal pairs of elements with one prescribed trace

Let $q, n, m \in \mathbb{N}$ such that $q$ is a prime power, $m \geq 3$ and $a \in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($α$, $f(α)$) in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ such that Tr$_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(α^{-1})=a$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with degree sum $n$. In particular, for $q=5^k, ~k \geq 5$ and degree sum $n=4$, we explicitly find at most 11 choices of $(q, m)$ where existence of such pairs is not guaranteed.

math.NT↗
arXiv2022

Characteristic functions for \MakeLowercase{(r, n)}-free and \MakeLowercase{(f, g)}-free elements

For a prime power $q$, $\F$ denotes the finite field of order $q$, and for $m\geq 2$, $\Fm$ denotes the extension field of degree $m$. We establish a characteristic function for the set of $(r,\, n)$-free elements of finite cyclic $R$-module for the Euclidean domain $R$. Furthermore, we explore $(f,\, g)$-freeness through polynomial values and finally give an expression for the characteristic function for the set of $(f,\, g)$-free elements.

math.NT↗
arXiv2022

Pair of primitive elements in quadratic form with prescribed trace over a finite field

In this article, we establish a sufficient condition for the existence of primitive element $α\in \Fm$ is such that $f(α)$ is also primitive element of $\Fm$ and $Tr_{\Fm/\F}(α)=β$, for any prescribed $β\in\F$, where $f(x)= ax^2 + bx + c\in \Fm(x)$ such that $b^2-4ac\neq 0$. We conclude that, for $m\geq 5$ there is only one exceptional pair $(q,m)$ which is $(2,6)$.

math.RA↗
arXiv2020

The existence of primitive normal elements of quadratic forms over finite fields

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive normal element $α$ such that $f(α)$ is a primitive element, where $f(x)= ax^2+bx+c$, with $a,b,c\in \mathbb{F}_{q^m}$ satisfying $b^2\neq ac$ in $\Fm$ except for at most 9 exceptional pairs $(q,m)$.

math.NT↗