相关论文: Reciprocal cyclotomic polynomials

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime…

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas…

We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes,…

We present an elementary identity for the cyclotomic polynomials $\Phi_n(X)$ which reflects a kind of multiplicative property of $\Phi_n(X)$ as a function of $n$, and we explore its connections with the properties of other arithmetical…

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this…

In this article, we provide a short and elementary proof of the following result: For $n \geq 3$ the middle coefficient of $\Phi_n(x)$ is either zero (when $n$ is a power of $2$) or an odd integer.

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em…

Given two cyclotomic polynomials $\Phi_n(x)$ and $\Phi_m(x)$, $n\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)\Phi_n (x)+b(x)\Phi_m(x),$$ with a(x) and b(x) integer polynomials.

Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we…

Let $N=p_1p_2... p_n$ be a product of $n$ distinct primes. Define $P_N(x)$ to be the polynomial $(1-x^N)\prod_{1\leq i<j\leq n}(1-x^{N/(p_ip_j)})/\prod_{i=1}^n (1-x^{N/p_i})$. (When $n=2$, $P_{pq}(x)$ is the $pq$-th cyclotomic polynomial,…

We introduce and study the generalized cyclotomic polynomials $\Phi_{A,S,n}(x)$ associated with a regular system $A$ of divisors and an arbitrary set $S$ of positive integers. We show that all of these polynomials have integer coefficients,…

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the…

We present several approaches on finding necessary and sufficient conditions on $n$ so that $\Phi_k(x^n)$ is irreducible where $\Phi_k$ is the $k$-th cyclotomic polynomial.

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…

Let $\Phi_n(X)$ and $\Psi_n(X)=\frac{X^{n}-1}{\Phi_{n}(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $ d_{1} \neq d_{2} $ of $ n $, and integers $l_1$ and $l_2$…

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$,…

A cyclotomic polynomial \Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we…

Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of…