Cyclotomic coefficients: gaps and jumps

Abstract

We improve several recent results by Hong, Lee, Lee and Park (2012) on gaps and Bzdȩga (2014) on jumps amongst the coefficients of cyclotomic polynomials. Besides direct improvements, we also introduce several new techniques that have never been used in this area.

Binary Cyclotomic Polynomials: Representation via Words and Algorithms

“…The fact that g(F p,q ) = p − 1 was proven in [3] for binary cyclotomic polynomials Φ pq and in [6] for semi-group polynomials. In [2,7], it is proven that the number of maximum gaps in binary cyclotomic polynomials is 2⌊q/p⌋. Our description in terms of words easily entails the inequality g(F p,q ) p − 1 and proves that there are, at least, 2⌊q/p⌋ maximum gaps.…”

Products of cyclotomic polynomials on unit circle

“…of jumps of ternary cyclotomic coefficients studied by the author [3] and Camburu, Ciolan, Luca, Moree and Shparlinski [5].…”

A Survey on Coefficients of Cyclotomic Polynomials

“…Yet another short proof was given by Kaplan (2016) [30, End of Section 2.1]. Furthermore, Camburu, Ciolan, Luca, Moree, and Shparlinski (2016) [30] determined the number of maximum gaps of Φ pq (X), and the existence of particular gaps in the case in which q ≡ ±1 (mod p). The following theorem collects these results [30,58,88].…”

“…Bzdȩga [26] proved that J n > n 1/3 for all ternary integers n. As a corollary, θ n > n 1/3 . Also, he showed that Schinzel Hypothesis H implies that for every ε > 0 we have J n < 10n 1/3+ε for infinitely many ternary integers n. Camburu, Ciolan, Luca, Moree, and Shparlinski (2016) [30] gave an unconditional proof that J n < n 7/8+ε for infinitely many ternary integers n.…”