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Coefficients of a relative of cyclotomic polynomials

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Liu R.

Acta Arithmetica

2014

tom 165

nr 4

301-325

Let $N = p_1 ⋯ p_n$ be a product of n ≥ distinct primes. Define $P_N(x)$ to be the polynomial
$(1-x^N) ∏_{1≤i(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 $2^{n^2/2+O(n logn)}$.

Ograniczanie wyników

1 Acta Arithmetica

1 Liu R.

1 2014

Wyniki wyszukiwania

Coefficients of a relative of cyclotomic polynomials