Coefficients of a relative of cyclotomic polynomials

TL;DR

This paper investigates the coefficients of a polynomial family related to cyclotomic polynomials, revealing their dependence on residue orderings and establishing bounds on their maximum coefficient magnitude.

Contribution

It explicitly describes coefficients for the case n=3 and bounds the polynomial height for n=4, also analyzing growth of maximum height for general n.

Findings

Coefficients depend mainly on the order of sums of residues.

Explicit description of coefficients when n=3.

Maximum height is at most 2 for n=4.

Abstract

Let $N=p_1p_2... p_n$ be a product of $n$ 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, 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 one of its coefficients. It is well known that the height of $\Phi_{pq}(x)$ 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 exist $P_N(x)$ with height 1 but that in general the maximum height of $P_N(x)$ is a function depending only on $n$ with growth rate $2^{n^2/2+O(n\log n)}$.