Related papers: Minimally Intersective Polynomials with Arbitraril…

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$)…

Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\…

In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all $d\geq 2$, there exists a polynomial $f_d(x) \in \mathbb{Q}[x]$ with $2\leq \mathrm{deg}(f_d)…

Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P :=\prod_{n \geq 0} (\frac{2n+1}{2n+2})^{(-1)^{N_{1,2}(n)}}$ and Robbins proved that $P = 1/\sqrt{2}$. Related…

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…

The partition function, $p_A(n)$, is defined to be the number of partitions of $n$ with parts in the set A, where $n$ is a positive integer and $A$ is a set of positive integers. It is well documented that: if A is a finite set with…

Let $z\ne \pm1,w^2$ be a fixed integer, and let $f(t)\ne g(t)^2$ be a fixed polynomial over the integers. It is shown that the subset of primes $p\geq 2$ such that $z$ and $f(z)$ is a pair of simultaneous primitive roots modulo $p$ has…

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278 is…

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…

We define an Artin prime for an integer $g$ to be a prime such that $g$ is a primitive root modulo that prime. Let $g\in \mathbb{Z}\setminus\{-1\}$ and not be a perfect square. A conjecture of Artin states that the set of Artin primes for…

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that…

The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having…

For any $\epsilon>0$, there exists $q_0(\epsilon)$ such for any $q\ge q_0(\epsilon)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three…

Let $k$ be an arbitrary given positive integer and let $f(x)\in {\mathbb Z}[x]$ be a quadratic polynomial with $a$ and $D$ as its leading coefficient and discriminant, respectively. Associated to the least common multiple ${\rm lcm}_{0\le…

We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we…

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…

We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these…

Given a graph $G$ of order $n$, the $\sigma$-$polynomial$ of $G$ is the generating function $\sigma(G,x) = \sum a_{i}x^{i}$ where $a_{i}$ is the number of partitions of the vertex set of $G$ into $i$ nonempty independent sets. Such…

Let $f_1,\dots,f_k \in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an $n<x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on an earlier…

We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…