Related papers: On bi-unitary perfect polynomials over $F_2$

In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into…

It is very fundamental to study irreducible plane curve singularities in algebraic geometry. The contents of the paper consist of three parts, called Part[A], Part[B] and Part[C] with Good Appendix. Our aim is to prove by Part[B] and…

Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these…

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready…

In this article we study the irreducibility of polynomials of the form $x^n+\epsilon_1 x^m+p^k\epsilon_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and…

In this paper we give a different proof of Kuz'min's result on the number of irreducible polynomials with the first two coefficients fixed. Our technique is to relate the question to the number of points on a curve, and to calculate the…

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of…

This paper addresses the factorization of polynomials of the form $F(x) = f_{0}(x) + f_{1}(x) x^{n} + \cdots + f_{r-1}(x) x^{(r-1)n} + f_{r}(x) x^{rn}$ where $r$ is a fixed positive integer and the $f_{j}(x)$ are fixed polynomials in…

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is…

We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on…

Let $p$ be a prime. We classify the finite groups having exactly two irreducible $p$-Brauer characters of degree larger than one. The case, where the finite groups have orders not divisible by $p$, was done by P. P\'alfy in 1981.

This paper investigates whether or not polynomials that are irreducible over $\mathbb{Q}$ and $\mathbb{Z}$ can remain irreducible under substitution by all quadratic polynomials. It answers this question in the negative in the degree 2 and…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of…

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…

We prove that, over a field of characteristic $0$, the degrees of factors of a binomial $t^n-\alpha$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the…

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their…