Resultants of dynatomic polynomials of xd
Abstract
Let K be a field of characteristic zero, and let ϕ(x)∈ K[x] be a polynomial of degree at least 2. Denote the n-th iterate of ϕ by ϕn. The n-th dynatomic polynomial of ϕ is defined by Φϕ,n(x) := ∏k| n(ϕk(x)−x)μ(n/k). In this paper, we specialize to the case ϕ(x) = xd. We first establish several properties of Φϕ,n that are analogous to those of cyclotomic polynomials. We then combine these properties with known results on resultants of cyclotomic polynomials to determine the resultants of dynatomic polynomials. In particular, for 1≤ n< m, we show that Res(Φϕ,n,Φϕ,m) = 1 if and only if nnmid m. When n| m, we obtain the explicit formula Res(Φϕ,n,Φϕ,m) = Φm/n(dn)νd(n), where Φj is the j-th cyclotomic polynomial, νd(1) = d−1, and νd(n) = degΦϕ,n for n>1.
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BibTeX
@misc{kao2026resultants,
title = {{Resultants of dynatomic polynomials of $x^d$}},
author = {Chih-Chiang Kao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27016},
eprint = {2608.27016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27016},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Resultants of dynatomic polynomials of x^d
AU - Chih-Chiang Kao
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.27016
UR - https://arxiv.org/abs/2608.27016
AB - Let K be a field of characteristic zero, and let ϕ(x)∈ K[x] be a polynomial of degree at least 2. Denote the n-th iterate of ϕ by ϕ^n. The n-th dynatomic polynomial of ϕ is defined by Φ_ϕ,n(x) := ∏_k| n(ϕ^k(x)−x)^μ(n/k). In this paper, we specialize to the case ϕ(x) = x^d. We first establish several properties of Φ_ϕ,n that are analogous to those of cyclotomic polynomials. We then combine these properties with known results on resultants of cyclotomic polynomials to determine the resultants of dynatomic polynomials. In particular, for 1≤ n< m, we show that Res(Φ_ϕ,n,Φ_ϕ,m) = 1 if and only if nnmid m. When n| m, we obtain the explicit formula Res(Φ_ϕ,n,Φ_ϕ,m) = Φ_m/n(d^n)^ν_d(n), where Φ_j is the j-th cyclotomic polynomial, ν_d(1) = d−1, and ν_d(n) = degΦ_ϕ,n for n>1.
ER -