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CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 1 / August 2019
- Published online by Cambridge University Press:
- 28 March 2019, pp. 41-47
- Print publication:
- August 2019
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- Article
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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’, Integers12(4) (2012), 503–519].
