Ternary cyclotomic polynomials with an optimally large set of coefficients

Bachman, Gennady

doi:10.1090/S0002-9939-04-07338-1

Ternary cyclotomic polynomials with an optimally large set of coefficients
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Proc. Amer. Math. Soc. 132 (2004), 1943-1950 Request permission

Abstract:

Ternary cyclotomic polynomials are polynomials of the form $\Phi _{pqr}(z)=\prod _\rho (z-\rho )$, where $p<q<r$ are odd primes and the product is taken over all primitive $pqr$-th roots of unity $\rho$. We show that for every $p$ there exists an infinite family of polynomials $\Phi _{pqr}$ such that the set of coefficients of each of these polynomials coincides with the set of integers in the interval $[-(p-1)/2,(p+1)/2]$. It is known that no larger range is possible even if gaps in the range are permitted.

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