The Height of a Class of Ternary Cyclotomic Polynomials

Abstract

Abstract. Let A(n) denote the largest absolute value of the coefficients of n-th cyclotomic polynomial Φn(x) and let p < q < r be odd primes. In this note, we give an infinite family of cyclotomic polynomials Φpqr(x) with A(pqr) = 3, without fixing p.

A Survey on Coefficients of Cyclotomic Polynomials

“…Kaplan (2007) [64, Lemma 1] proved the following lemma, which provides a formula for the coefficients of ternary cyclotomic polynomials. This is known as Kaplan's lemma and has been used to prove several results on ternary cyclotomic polynomials [37,48,49,51,60,89,100,101,102,105,106]. Lemma 3.1 (Kaplan's lemma).…”

“…Furthermore, for q ≡ 1 (mod p) and r ≡ −2 (mod pq), Zhang (2014) [100] constructed an explicit j such that a pqr (j) = −2, so that Φ pqr (X) is not flat. Regarding nonflat ternary cyclotomic polynomials with small heights, Zhang (2017) [101] showed that for every prime p ≡ 1 (mod 3) there exist infinitely many q and r such that A(pqr) = 3.…”