Bin Zhang: A remark on bounds for the maximal height of divisors of $x^n-1$, 209-214

Abstract:

For a given positive integer $n$, let $B(n)$ denote the maximum absolute value of coefficients of any divisor of $x^n-1$. Let $r\geq2$ and $n=p_1^{e_1}\cdots p_r^{e_r}$, where $p_1<\cdots<p_r$ are distinct primes and each $e_i>0$. In this paper, we show that

$$B(n)<(\frac{2}{5})^{\prod\limits_{i=2}^re_i}\prod_{i=1}^{r-1}p_i^{4\cdot3^{r-2}E-e_i},$$

where $E=\prod_{j=1}^re_j$. This generalizes a result of Kaplan [J. Number Theory, 129(2009), pp. 2673-2688].

Key Words: Cyclotomic polynomial, heights of polynomials, widths of polynomials.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 11C08.