Ternary cyclotomic polynomials with an optimally large set of coefficients

Abstract

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