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Bartlomiej Bzdega

Publications and source records attributed to Bartlomiej Bzdega.

10 recordsLinked to original sources
arXiv2017

Cyclotomic polynomials at roots of unity

The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $Φ_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $Φ_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $Φ_n(e^{2πi/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $Φ_n(x)$. We also obtain a formula for $Φ_n'(e^{2πi/m}) / Φ_n(e^{2πi/m})$ with $n \ne m$, which is effectively applied to $m \in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.

math.NT↗
arXiv2016

Products of cyclotomic polynomials on unit circle

We present a method to deal with the values of polynomials of type $P(z)=\prod_{d\in D}(1-z^d)^{j_d}$ on the unit circle. We use it to improve the known bounds on various measures of coefficients of cyclotomic and similar polynomials.

math.NT↗
arXiv2014

On a generalization of Beiter Conjecture

We prove that for every $\varepsilon>0$ and a nonnegative integer $ω$ there exist primes $p_1,p_2,\ldots,p_ω$ such that for $n=p_1p_2\ldots p_ω$ the height of the cyclotomic polynomial $Φ_n$ is at least $(1-\varepsilon)c_ωM_n$, where $M_n=\prod_{i=1}^{ω-2}p_i^{2^{ω-1-i}-1}$ and $c_ω$ is a constant depending only on $ω$; furthermore $\lim_{ω\to\infty}c_ω^{2^{-ω}}\approx0.71$. In our construction we can have $p_i>h(p_1p_2\ldots p_{i-1})$ for all $i=1,2,\ldots,ω$ and any function $h:\mathbb{R}_+\to\mathbb{R}_+$.

math.NT↗
arXiv2013

On a certain family of inverse ternary cyclotomic polynomials

We study a family of inverse ternary cyclotomic polynomials $Ψ_{pqr}$ in which $r\leφ(pq)$ is a positive linear combination of $p$ and $q$. We derive a formula for the height of such polynomial and characterize all flat polynomials in this family.

math.NT↗
arXiv2013

Jumps of ternary cyclotomic coefficients

It is known that two consecutive coefficients of a ternary cyclotomic polynomial $Φ_{pqr}(x)=\sum_k a_{pqr}(k)x^k$ differ by at most one. In this paper we give a criterion on $k$ to satisfy $|a_{pqr}(k)-a_{pqr}(k-1)|=1$. We use this to prove that the number of nonzero coefficients of the $n$th ternary cyclotomic polynomial is greater than $n^{1/3}$.

math.NT↗
arXiv2012

Inclusion-exclusion polynomials with large coefficients

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 $q_1<q_2<...<q_k$ are pairwise coprime and arbitrary large.

math.NT↗
arXiv2010

On the height of cyclotomic polynomials

Let $A_n$ denote the height of cyclotomic polynomial $Φ_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le ε_kϕ(n)^{k^{-1}2^{k-1}-1}$ with $-\logε_k\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $Ψ_n$. Additionally, we improve on a bound of Kaplan for the maximal height of divisors of $x^n-1$, denoted by $B_n$. We show that $B_n<η_k n^{(3^k-1)/(2k)-1}$, with $-\log η_k \sim c3^k$ and the same $c$.

math.NT↗
arXiv2008

Bounds on ternary cyclotomic coefficients

We present a new bound on $A = \max_n |a_{pqr}(n)|$, where $a_{pqr}(n)$ are the coefficients of a ternary cyclotomic polynomial. We also prove that two consecutive coefficients of such a polynomial differ by at most one.

math.NT↗