Related papers: Sums of polynomial-type exceptional units modulo $…

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is…

Let $G$ be a finite cyclic group, written additively, and let $A,\ B$ be nonempty subsets of $G$. We will say that $G= A+B$ is a \textit{factorization} if for each $g$ in $G$ there are unique elements $a,\ b$ of $G$ such that $g=a+b, \ a\in…

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log\log x)^{1/835}$ consecutive…

In this paper, we compute the size of the exceptional set in a generalized Goldbach problem and show that for a given polynomial $f(x) \in \mathbb{Z}[x]$ with a positive leading coefficient, positive integers $A$, $B$, $g$ and $0 \leq i, j…

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 integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant…

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over the finite field $\mathbb{F}_q$. Building off of techniques of Balog and Ruzsa and of Keil in the integer setting, we determine the precise order of magnitude of $k$th moments of…

We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for…

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $k+m$ and a non-negative integer $r$, count the number of…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\sum_{k=r(mod p^{\beta})}\binom{n}{k}(-1)^k f([(k-r)/p^{\alpha}]),$$…

We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the…

We investigate the existence of maximal collections of mutually noncrossing $k$-element subsets of $\left\{ 1, \dots, n \right\}$ that are invariant under adding $k\pmod n$ to all indices. Our main result is that such a collection exists if…

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. For all integers $a$ and $b \geq 1$ with $\gcd(a, b) = 1$, let $[a^{-1} \!\bmod b]$ be the multiplicative inverse of $a$ modulo $b$, which we pick in the usual set of…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

Let $f(x)=(x^{k}+c)^{m}-ax^{n}\in\mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$, where $k,m,n\in\mathbb{N}$ with $km>n$, and let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $f(x)$. We investigate the arithmetic…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…

Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta$ be a root of $f(x)$, set $K= \mathbb{Q}(\theta)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted…

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…