Related papers: Sequences of binary irreducible polynomials

While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq…

In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

A second order polynomial sequence is of Fibonacci-type $\mathcal{F}_{n}$ (Lucas-type $\mathcal{L}_{n}$) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Under certain conditions these polynomials are…

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…

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting…

A-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field F_2, since they constitute the starting point of the iteration. The exact…

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…

The goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\ldots, 210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove…

Irreducible trinomials of given degree n over $F_2$ do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we…

In this article, we give a family of examples of algebras, showing that for every $n \geq 2$ and $m \geq 0$, there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the…

We study the number of irreducible polynomials over $\mathbf{F}_{q}$ with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree $n$ with $r$ coefficients…

In this paper, we provide the degree distribution of irreducible factors of the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial. We…

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…

For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in…

Let ${\mathcal B}=\{b_i \}_{i=1}^\infty$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\leq s \leq r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic…

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 $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct…

Let $n (>3)$ be a prime number and $\Bbb F_{2^n}$ a finite field of $2^n$ elements. Let $L =\Bbb F_{2^n}\cup \{\infty\}$ be the support set and $g(x)$ an irreducible polynomial of degree $6$ over $\Bbb F_{2^n}$. In this paper, we obtain an…

Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…