Year: 2026
Type: preprint
Abstract: Let $K$ be a field of characteristic zero, and let $ϕ(x)\in K[x]$ 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 $$Φ_{ϕ,n}(x) := \prod_{k\mid n}(ϕ^k(x)-x)^{μ(n/k)}.$$ 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 $1\leq n< m$, we show that $\operatorname{Res}(Φ_{ϕ,n},Φ_{ϕ,m}) = 1$ if and only if $n\nmid m$. When $n\mid m$, we obtain the explicit formula $$\operatorname{Res}(Φ_{ϕ,n},Φ_{ϕ,m}) = Φ_{m/n}(d^n)^{ν_d(n)},$$ where $Φ_j$ is the $j$-th cyclotomic polynomial, $ν_d(1) = d-1$, and $ν_d(n) = \operatorname{deg}Φ_{ϕ,n}$ for $n>1$.
Source: arXiv (Cornell University)
Author: Kao, Chih-Chiang
Institution: Institute of Science Tokyo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204547706