Journal ArticleDOI
Cyclotomic factors of Borwein polynomials
TLDR
In this article, it was shown that if a monic polynomial with coefficients from the unit circle has a cyclotomic factor and every prime divisor of the polynomials is less than or equal to the number of terms of the terms, then it has an essential cyclotome factor.Abstract:
A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers 12(4) (2012), 503–519].read more
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Journal ArticleDOI
On Vanishing Sums of Roots of Unity
Tsit Yuen Lam,Ka Hin Leung +1 more
TL;DR: In this article, it was shown that the set of all possible n's is exactly the collection of N -combinations of the prime divisors of a given natural number m, where n denotes the subset of all non-negative integers in m. The proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity.
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On vanishing sums for roots of unity
T. Y. Lam,K. H. Leung +1 more
TL;DR: In this article, it was shown that the answer is exactly the set of linear combinations with non-negative integer coefficients of the prime factors of the root of unity in the Euclidean space.