Coefficients of a relative of cyclotomic polynomials
Volume 165 / 2014
Abstract
Let $N=p_1\cdots p_n$ be a product of $n \geq 2$ distinct primes. Define $P_N(x)$ to be the polynomial $$ (1-x^N)\prod_{1\leq i< j\leq n}(1-x^{N/(p_ip_j)})/\prod_{i=1}^n (1-x^{N/p_i}). $$ (When $n=2$, $P_{pq}(x)$ is the $pq$th cyclotomic polynomial $\varPhi_{pq}(x)$, and when $n=3$, $P_{pqr}(x)$ is $1-x$ times the $pqr$th cyclotomic polynomial.) Let the height of a polynomial be the maximum absolute value of its coefficients. It is well known that the height of $\varPhi_{pq}(x)$ is 1, and Gallot and Moree showed that the same is true for $P_{pqr}(x)$ when $n=3$. We show that the coefficients of $P_N(x)$ depend mainly on the relative order of sums of residues of the form $p_j^{-1} \pmod {p_i}$. This allows us to explicitly describe the coefficients of $P_N(x)$ when $n=3$ and show that the height of $P_N(x)$ is at most 2 when $n=4$. We also show that for any $n$ there exists $P_N(x)$ with height 1 but that in general the maximum height of $P_N(x)$ is a function depending only on $n$ with growth rate $2^{n^2/2+O(n\log n)}$.
