发表机构
Institute of Science Tokyo（东京科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对φ(x)=x^d的特殊情形，研究动力多项式的结式，建立其与分圆多项式类似的性质，证明1≤n<m时Res(Φ_{φ,n},Φ_{φ,m})=1当且仅当n不整除m，并得到Res(Φ_{φ,1},Φ_{φ,m})的显式公式。
AI 中文摘要
设K为特征零的域，φ(x)∈K[x]为次数至少为2的多项式，记φ的第n次迭代为φⁿ，φ的第n个动力多项式定义为Φ_{φ,n}(x):=∏_{k|n}(φᵏ(x)-x)^(μ(n/k))。本文专门研究φ(x)=x^d的情形，首先建立Φ_{φ,n}的若干与分圆多项式类似的性质，再将这些性质与分圆多项式结式的已知结果结合以确定动力多项式的结式。特别地，对于1≤n<m，证明Res(Φ_{φ,n},Φ_{φ,m})=1当且仅当n不整除m，且得到Res(Φ_{φ,1},Φ_{φ,m})的显式公式。
英文摘要
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 $n>1$.
Comments10 pages
