Related papers: More Classes of Complete Permutation Polynomials o…
Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is a important measure of complexity of the polynomial. In this paper we find the sharp lower bound for the weight of any permutation…
In this article we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two variable Kampe de Feriet family. We…
Recently, Boole polynomials have been studied by Kim and Kim over the p-adic number field. In this paper, we consider a q-extension of Boole polynomials by using the fermionic p-adic integrals on Zp and give some new identities related to…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing…
Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…
Let $f={\tt X}^r(a+{\tt X}^{2(q-1)})\in{\Bbb F}_{q^2}[{\tt X}]$, where $a\in{\Bbb F}_{q^2}^*$ and $r\ge 1$. The parameters $(q,r,a)$ for which $f$ is a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ have been determined in the following…
In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials.
In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…
We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials $f_{\alpha,\beta}(x)= x + \alpha x^{q(q-1)+1} + \beta x^{2(q-1)+1} \in \mathbb{F}_{q^2}[x]$, $\alpha\beta \neq 0$, $q$ even, characterizing all the pairs…
The paper is about an arithmetic problem in $\F_2[x]$. We give \emph{admissible} (necessary) conditions satisfied by a set of odd prime divisors of perfect polynomials over $\F_2$. This allows us to prove a new characterization of…
In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length $n$ over $\mathbb{F}_q$ and $\mathbb{Z}_k$. We give the MacWilliams type identity for the complete joint Jacobi polynomials of…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
This article introduces an analogue of permutation classes in the context of polyominoes. For both permutation classes and polyomino classes, we present an original way of characterizing them by avoidance constraints (namely, with excluded…
We show that the effective factorization of Ore polynomials over $\mathbb{F}_q(t)$ is still an open problem. This is so because the known algorithm in [1] presents two gaps, and therefore it does not cover all the examples. We amend one of…
We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes…
The previous paper [4] proved the existence of primitive polynomials and primitive normal polynomials of degree n with k prescribed coefficients in the finite field GF(q) for all sufficiently large q. This paper presents a loger versions of…
The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…
A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…
We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the…