arXiv subjects
Publications and source records attributed to Dhiren Kumar Basnet.
In this article, we establish a bound on $\theta_q$ that guarantees the existence of a pair of consecutive non-primitive elements in $\mathbb{F}_q$, with the exceptions $q=4$ and $q=8$. We first derive a sufficient condition for the existence of such a pair using character sums and then obtain the stated bound by considering several cases according to the least prime divisor of $q-1$.
In this article, we study polynomials over $\mathbb{F}_{q^2}$ that do not permute $\mathbb{F}_{q^2}$. More precisely, we characterize polynomials of the forms $x^q + b x^2 + c x + d$ and $x^{q+1} + b x^q + c x + d$ over $\mathbb{F}_{q^2}$ according to whether they are permutation or non-permutation polynomials. To this end, we determine the exact number of zeros of these polynomials using existing results on certain special Weil sums.
Let $q$ be a prime power and $m>1$ be any integer. Let $\mathbb F_{q^m}$ be the finite field of order $q^m$ and $\theta\in\mathbb F_{q^m}$ be such that $\mathbb F_{q^m} = \mathbb F(\theta)$. We obtain a nontrivial bound for the mixed character sum $\sum_{x \in\mathbb F}\chi(\theta+x)\psi(x)$, where $\chi$ and $\psi$ are multiplicative and additive characters of $\mathbb F_{q^m}$ and $\mathbb F$, respectively, using function field methods. As an application of our main result, we prove that for fixed $m$ and sufficiently large prime powers $q$, that satisfy certain conditions, $\mathbb F_{q^m}/\mathbb F$ possesses the weak line property for primitive normal elements. In particular, our result is a strengthening of existing results.
In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.
In 2019, Xiang Fan \cite{xfan} classified all permutation polynomials of degree $7$ over finite fields of odd characteristics. In this paper, we use this classification to determine the complete list of degree $7$ orthomorphism polynomials over finite fields of order $q\in\{11,~13,~17,~19,~25,~49\}.$ In addition, the non-existence of these polynomials is established for certain fields.
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 $\alpha$ of $\Fm$, such that $f(\alpha)$ 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 $(\alpha, f(\alpha))$ avoids each $A_i$. We establish results for fields of higher order.
Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $\epsilon \in \mathbb{F}_{q^m}$, such that $(f(\epsilon), g(\epsilon))$ is a primitive pair in $\mathbb{F}_{q^m}$ with two prescribed traces, $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a$ and $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon^{-1})=b$, where $f(x), g(x) \in \mathbb{F}_{q^m}(x)$ are rational functions with some restrictions and $a, b \in \mathbb{F}_q$. Also, we show that there exists an element $\epsilon \in \mathbb{F}_{q^m}$ satisfying our desired properties in all but finitely many fields $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$. We also calculate possible exceptional pairs explicitly for $m\geq 9$, when degree sums of both the rational functions are taken to be 3.
We know that if there are $k$ distinct prime factors of $n \in \mathbb{N}$, then the ring $\mathbb{Z}_n$ of integers modulo $n$ has exactly $2^k$ idempotent elements. In this article, we try to describe all the idempotents of $\mathbb{Z}_n$ for any given $n \in \mathbb{N}$.
In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over $\mathbb{F}_{q^{2}}$. The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in $μ_{q+1}$ (the set of $(q+1)-\emph{th}$ roots of unity), which helps in the determination of polynomials that permute $\mathbb{F}_{q^{2}}$. In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type $ax^{q+1}+bx^{q}+bx+a^{q}$ taking both the cases under consideration viz. $a\in \mathbb{F}_{q}$ and $a\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ both.
Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of primitive normal pairs of the type $(\epsilon, f(\epsilon))$ in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ with two prescribed traces, $Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a$ and $Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with some restrictions and $a, b \in \mathbb{F}_q$. Furthermore, for $q=5^k$, $m \geq 9$ and rational functions with degree sum 4, we explicitly find at most 12 fields in which the desired pair may not exist.
Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(α, f(α)) \in \mathbb{F}_{q^m}$ over $\mathbb{F}$ with a prescribed primitive norm $a$ and a non-zero trace $b$ over $\mathbb{F}$ of $α$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$ with some minor restrictions. Furthermore, for $q=7^k$, $m \geq 7$ and rational functions with numerator and denominator being linear, we explicitly find at most 6 fields in which the desired pair may not exist.
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 ($\alpha$, $f(\alpha)$) in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ such that Tr$_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(\alpha^{-1})=a$, 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.
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.
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 $(2,6)$.
We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.
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 $(\alpha, \, f(\alpha))$, 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 $(q,m)$.
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 $(q,m)$.
In this article, we introduce the concept of weakly $I$-clean ring, for any ideal $I$ of a ring $R$. We show that, for an ideal $I$ of a ring $R$, $R$ is uniquely weakly $I$-clean if and only if $R/I$ is semi boolean and idempotents can be lifted uniquely weakly modulo $I$ if and only if for each $a\in R$, there exists a central idempotent $e\in R$ such that either $a-e\in I$ or $a+e\in I$ and $I$ is idempotent free. As a corollary, we characterize weakly $J$-clean ring. Also we study various properties of weakly $I$-clean ring.