Related papers: A study of some recent irreducibility criteria for…
H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the…
Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect…
We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the…
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…
In this paper, we introduce two primality tests based on new divisibility properties of binomial coefficients. These new properties were enunciated and proved in previous work. We also study two similar tests that can be obtained from…
Expansive polynomials (whose roots are greater than 1 in modulus) often arise in dynamical systems and other computational problems. This paper examines the expansivity gap (the gap between 1 and the smallest modulus of the roots) of these…
In this note, we prove an irreducibility criterion for the polynomial of the form $f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{m}x^{m} + p^{u} \in \mathbb{Z}[x]$, where $p$ is a prime number, $u \geqslant 1$, $\gcd(u, m) = 1$, $p \nmid…
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…
We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…
For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results…
We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.
In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence…
Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…
Motivated by the question of whether a random polynomial with integer coefficients is likely to be irreducible, we study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is…
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…
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…
We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator,…
We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$…