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Acta Arithmetica

2014 | 165 | 4 | 301-325

Coefficients of a relative of cyclotomic polynomials

EN

Abstrakty

EN
Let $N = p_1 ⋯ p_n$ be a product of n ≥ distinct primes. Define $P_N(x)$ to be the polynomial <br> $(1-x^N) ∏_{1≤i<j≤n} (1-x^{N/(p_ip_j)})/ ∏_{i=1}^n (1-x^{N/p_i})$. <br>(When n=2, $P_{pq}(x)$ is the pqth cyclotomic polynomial $Φ_{pq}(x)$, and when n=3, $P_{pqr}(x)$ is 1-x times the pqrth cyclotomic polynomial.) Let the height of a polynomial be the maximum absolute value of its coefficients. It is well known that the height of $Φ_{pq}(x)$ 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 $p_j^{-1} (mod p_i)$. 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 $2^{n^2/2+O(n logn)}$.

301-325

wydano
2014

Twórcy

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