Coefficients of a relative of cyclotomic polynomials

Ricky Ini Liu¹

¹ Department of Mathematics University of Michigan Ann Arbor, MI 48109, U.S.A.

Acta Arithmetica, Tome 165 (2014) no. 4, pp. 301-325

Résumé

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 $2^{n^2/2+O(n\log n)}$.

Détail

DOI : 10.4064/aa165-4-1

Mots-clés : cdots product geq distinct primes define polynomial x prod leq leq x prod x pqth cyclotomic polynomial varphi pqr x times pqrth cyclotomic polynomial height polynomial maximum absolute value its coefficients known height varphi gallot moree showed pqr coefficients depend mainly relative order sums residues form pmod allows explicitly describe coefficients height there exists height general maximum height function depending only growth rate log