Related papers: Revisiting Eisenstein-type criterion over integers

We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that…

We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the…

Let f be an irreducible polynomial of degree d>=3 with no fixed prime divisor. We derive an asymptotic formula for the number of primes p<x such that f(p) is (d-1)-free.

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…

We recast Euclid's proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's "topological" proof of the infinitude of prime numbers and show that our…

Very simple sufficient conditions for the irreducibility of $f(X^n)$ over an arbitrary unique factorization domain $Z$ are established via a generalization of a well known theorem of A. Capelli.

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a…

Previously Heyman and Shparlinski gave an asymptotic formula with error term for the number of Eisenstein polynomials of fixed degree and bounded height. Let $\psi(f)$ denote the number of primes for which a polynomial $f$ is Eisenstein. We…

For an element $a$ of an integral domain D under an equivalence relation \tau, the \tau-factorization of a is defined as \lambda a_1 a_2... a_k, where \lambda is a unit in D and a_i \tau a_j for all i, j. An irreducible element has no…

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…

A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given by $y_0 = b$ and $y_{n+1} = f^{y_n}(a)$ with positive integers $a$ and $b$. We show…

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…

A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to one of two particular groups of small…

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…

Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.

We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the…

Towards a well-known open question in arithmetic dynamics, L. M\'erai, A. Ostafe and I. E. Shparlinski (2023), have shown, for a class of polynomials $f \in \mathbb Z[X]$, which in particular includes all quadratic polynomials, that, under…

The Schinzel Hypothesis is a celebrated conjecture in number theory linking polynomial values and prime numbers. In the same vein we investigate the common divisors of values $P_1(n),\ldots, P_s(n)$ of several polynomials. We deduce this…