Related papers: Irreducibility of Random Polynomials

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 study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f=y^n+\sum_{i=0}^{n-1}a_i(x)y^i\in\mathbb F_q[x][y]$ with i.i.d coefficients $a_i$ taking values in the set…

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) :…

We present a theorem about irreducibility of a polynomial that is the resultant of two others polynomials. The proof of this fact is based on the field theory. We also consider the converse theorem and some examples.

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

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…

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…

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary…

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…

Jakhar shown that for $f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdot+ a_0$ ($a_0\neq 0$) is a polynomial with rational coefficients, if there exists a prime integer $p$ satisfying $\nu_p(a_n)=0$ and $n\nu_p(a_i)\ge (n-i)\nu_p(a_0)> 0$ for every $0\le…

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…

If the product of two monic polynomials with real nonnegative coefficients has all coefficients equal to 0 or 1, does it follow that all the coefficients of the two factors are also equal to 0 or 1? Here is an equivalent formulation of this…

We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this…

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear…

In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are…

Let $p$ be a prime and $b(x)$ be an irreducible polynomial of degree $k$ over $\mathbb{F}_p$. Let $d\geq 1$ be an integer. Consider the following question: Is $b(x^d)$ irreducible? We derive necessary conditions for $b(x^d)$ to be…

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most…

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the…

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias",…

Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the field of fractions $\K$. Denote the ring of polynomials in $n$ variables over $\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by…