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Cyclotomic factors of Borwein polynomials. (English) Zbl 1432.11158

A Borwein polynomial is a polynomial \(f\) with coefficients in the set \(\{-1,0,1\}\). A cyclotomic factor \(\Phi_k(x)\) of \(f(x) \in \mathbb Z[x]\) is said to be essential if every prime divisor of \(k\) is less than or equal to the number of terms of \(f\). The authors show that if a Borwein polynomial has a cyclotomic factor then it has an essential cyclotomic factor. They use this to prove two conjectures of Mercer concerning the possible cyclotomic factors of polynomials that are the sum of 5 distinct powers of \(x\).

MSC:

11R09 Polynomials (irreducibility, etc.)
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References:

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