Coefficient Convexity of Divisors of $x^{n}-1$
DOI:
https://doi.org/10.5644/SJM.09.1.01Keywords:
Cyclotomic polynomials, coefficient sets of polynomialsAbstract
We say a polynomial $f\in {\mathbb Z}[x]$ is strongly coefficient convex if the set of coefficients of $f$ consists of consecutive integers only. We establish various results suggesting that the divisors of $x^n-1$ that are in ${\mathbb Z}[x]$ have the tendency to be strongly coefficient convex and have small coefficients. The case where $n=p^2q$ with $p$ and $q$ primes is studied in detail.
2010 Mathematics Subject Classification. 11B83, 11C08
Downloads
References
G. Bachman, On ternary inclusion-exclusion polynomials, Integers, 10 (2010), A48, 623-638.
B. Bzdęga, Bounds on ternary cyclotomic coefficients, Acta Arith., 144 (2010), 5-16.
L. Carlitz, The number of terms in the cyclotomic polynomial $Fsb{pq}(x)$}, Amer. Math. Monthly, 73 (1966), 979-981.
A. Decker and P. Moree, Coefficient convexity of divisors of $x^{n}-1$}, arXiv:1010.3938, pp. 34 (extended version of present paper).
Y. Gallot and P. Moree, Neighboring ternary cyclotomic coefficients differ by at most one, J. Ramanujan Math. Soc., 24 (2009), 235-248.
N. Kaplan, Bounds for the maximal height of divisors of $x^n-1$, J. Number Theory, 129 (2009), 2673--2688.
T.Y. Lam and K.H. Leung, On the cyclotomic polynomial $Phi_{pq}(X)$, Amer. Math. Monthly, 103 (1996), 562--564.
P. Moree, Inverse cyclotomic polynomials, J. Number Theory, 129 (2009), 667--680.
C. Pomerance and N. C. Ryan, Maximal height of divisors of $xsp n-1$, Illinois J. Math., 51 (2007), 597--604.
S. Rosset, The coefficients of cyclotomic like polynomials of order 3, unpublished manuscript (2008), pp. 5.
N.C. Ryan, B.C. Ward and R. Ward, Some conjectures on the maximal height of divisors of $x^n-1$, Involve 3 (2010), 451--457.
R. Thangadurai, On the coefficients of cyclotomic polynomials, Cyclotomic fields and related topics, (Pune, 1999), 311--322, Bhaskaracharya Pratishthana, Pune, 2000.
L. Thompson, Heights of divisors of $x^n-1$, Integers, 11A, Proceedings of the Integers Conference 2009 (2011), Article 20, 1--9.
