Related papers: A note regarding permutation binomials over finite…

We study the explicit factorization of $2^n r$-th cyclotomic polynomials over finite field $\mathbb{F}_q$ where $q, r$ are odd with $(r, q) =1$. We show that all irreducible factors of $2^n r$-th cyclotomic polynomials can be obtained…

In this paper we compute the exact divisibility of exponential sums associated to binomials $F(X)=aX^{d_1} +b X^{d_2}$. In particular, for the case where $\max\{d_1,d_2\}\leq\sqrt{p-1}$, the exact divisibility is computed. As a byproduct of…

Let $q$ be a prime power. We construct stable polynomials of the form $b^{m-1}(x+a)^m+c(x+a)+d$ over a finite field $\mathbb{F}_{q}$ for $m=2,3,4$ by Capelli's lemma. When $m=3$ and $q$ is even, we confirm the conjecture of Ahmadi and…

Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high…

We show that univariate trinomials $x^n + ax^s + b \in \mathbb{F}_q[x]$ can have at most $\delta \Big\lfloor \frac{1}{2} +\sqrt{\frac{q-1}{\delta}} \Big\rfloor$ distinct roots in $\mathbb{F}_q$, where $\delta = \gcd(n, s, q - 1)$. We also…

A well-known result of von zur Gathen asserts that a non-exceptional permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if $q<n^{4}$. With the help of the Weil bound for the number of $\mathbb{F}_{q}$-points on an…

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…

In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the…

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(\alpha)\mid\alpha\in\mathbb{F}_{q}\}$ and denote the…

In this paper we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+Ax^{q}+Bx)\in \mathbb{F}_{q^3}[x]$, with $A,B \in \mathbb{F}_{q}$. In particular we completely classify the pairs $(A,B)\in \mathbb{F}_{q}^2$ such that…

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree $8$ over $\mathbb{F}_{2^r}$ with $r>3$. By [J. Number Theory 176 (2017) 466-66], a polynomial $f$ of degree $8$ over $\mathbb{F}_{2^r}$ is…

Let $r$ be a prime power and $q=r^m$. For $0\le i\le m-1$, let $f_i\in \mathbb{F}_r[x]$ be $q$-linearized and $a_i\in \mathbb{F}_q$. Assume that $z\in \mathbb{\bar{F}}_r$ satisfies the equation $\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where…

An irreducible polynomial over $\Bbb F_q$ is said to be normal over $\Bbb F_q$ if its roots are linearly independent over $\Bbb F_q$. We show that there is a polynomial $h_n(X_1,\dots,X_n)\in\Bbb Z[X_1,\dots,X_n]$, independent of $q$, such…

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, and $T$ a positive integer. In this article we find a sharp estimative of the total number of monic irreducible binomials in $\mathbb F_q[x]$ of degree less or equal to $T$, when $T$…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

In this paper, we give error bounds on the number of monic irreducible polynomials $a_0+a_1x+\dots+a_{n-1}x^{n-1}+x^n$ over a finite field $\mathbb{F}_q$ of degree $n$ with $(a_0, a_1, \dots, a_{n-1}, 1)$ lying in a fixed affine algebraic…

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…