Related papers: Polynomials with divisors of every degree

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known…

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial…

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…

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows…

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…

We call a polynomial monogenic if a root $\theta$ has the property that $\mathbb{Z}[\theta]$ is the full ring of integers in $\mathbb{Q}(\theta)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any…

In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the…

We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…

We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\{q_1,q_2,...,q_k\}}$ with the height at least $c^{2^k}\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and…

We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log…

We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the…

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n -…

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ may be complex. We impose some restriction on the coefficients of the real part of the given polynomial…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…

For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…

We prove that there exists a universal constant $D$ such that if $p$ is a prime divisor of the index of the Fitting subgroup of a finite group $G$, then the number of conjugacy classes of G is at least $Dp/log_2 p$. We conjecture that we…

We give the classification, up to homeomorphisms, of reduced complex polynomials with 2 variables with one critical value.