Related papers: Reciprocal cyclotomic polynomials

In this paper I prove a conjecture which gives a lower bound for the largest absolute value of the coefficients of the n-th cyclotomic polynomial for some n. Moreover this estimate is essentially sharp.

Let $\Phi_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2\Phi^{(3)}_n(1)$ is divisible by $\phi(n)-2$ and $\Phi^{(2k+1)}_n(1)$ is divisible by…

This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let $k(n)$ denote the number of Kronecker…

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

We introduce a sequence $P_{2n}$ of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of $P_{2n}$ and its degree $d$ has a limit when…

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying…

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$.…

Let $n$ be an even positive integer with at most three distinct prime factors and let $\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \in \Q[x]$ having at least…

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the…

Let $f(Z)=Z^n-a_{1}Z^{n-1}+\cdots+(-1)^{n-1}a_{n-1}Z+(-1)^na_n$ be a monic polynomial with coefficients in a ring~$R$ with identity, not necessarily commutative. We study the ideal $I_f$ of $R[X_1,\dots,X_n]$ generated by…

Define the $n$-th fibotomic polynomial to be the product of the monic irredicible factors of the $n$-th Fibonacci polynomial which are not factors of any Fibonacci polynomial of smaller degree. In this paper, we prove a number of properties…

We study the number of real zeros of trigonometric polynomials in a period and the number of zeros of self-reciprocal algebraic polynomials on the unit circle under the assumption that their coefficients are in a fixed finite set of real…

We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.

I considered definition and properties of polynomial in no-commutative algebra. There exists polynomial which has finite, infinite or empty set of roots. For instance, the polynomial $$p_1(x)=ix-xi-1$$ have no root and the polynomial…

We investigate the polynomials $\sum_{k=0}^{n-1} c_n(k)x^k$ and $\sum_{k=0}^{n-1} |c_n(k)| x^k$, where $c_n(k)$ denote the Ramanujan sums. We point out connections and analogies to the cyclotomic polynomials.

We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial $f$ of degree $d$, there are exactly $2^{d-1}$ distinct degree $d$ polynomials with the same set of cyclic resultants as $f$.…

We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries…