Resultants of dynatomic polynomials of $x^d$

Abstract

Let $K$ be a field of characteristic zero, and let $\phi(x)\in K[x]$ be a polynomial of degree at least 2. Denote the $n$-th iterate of $\phi$ by $\phi^n$. The $n$-th dynatomic polynomial of $\phi$ is defined by $$\Phi_{\phi,n}(x) := \prod_{k\mid n}(\phi^k(x)-x)^{\mu(n/k)}.$$ In this paper, we specialize to the case $\phi(x) = x^d$. We first establish several properties of $\Phi_{\phi,n}$ 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 $\operatorname{Res}(\Phi_{\phi,n},\Phi_{\phi,m}) = 1$ if and only if $n\nmid m$, and we obtain an explicit formula for $\operatorname{Res}(\Phi_{\phi,1},\Phi_{\phi,m})$.