论文标题
关于有限反射组的作用下不变多项式的复杂性
On the complexity of invariant polynomials under the action of finite reflection groups
论文作者
论文摘要
令$ \ mathbb {k} [x_1，\ dots，x_n] $是字段上的多元多项式环$ \ mathbb {k} $。令$（u_1，\ dots，u_n）$为$ n $代数独立元素的序列，$ \ mathbb {k} [x_1，\ dots，x_n] $。给定一个多项式$ f $ in $ \ mathbb {k} [u_1，\ dots，u_n] $，$ \ mathbb {k} [x_1，\ dots，x_n] $的子来源是$ u_i $'s生成$ \ mathbb {k} [e_1，\ dots，e_n] $，其中$ e_1，\ dots，e_n $是新变量，使得$ f _ {\ mathrm {new}}}（u_1，u_1，\ dots，dots，dots，u_n）= f（x_1，\ _1，\ dots，x__n）$。我们提供算法并分析其算术复杂性，以计算$ f _ {\ mathrm {new}} $知道$ f $和$（u_1，\ dots，u_n）$。
Let $\mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$. Let $(u_1, \dots, u_n)$ be a sequence of $n$ algebraically independent elements in $\mathbb{K}[x_1, \dots, x_n]$. Given a polynomial $f$ in $\mathbb{K}[u_1, \dots, u_n]$, a subring of $\mathbb{K}[x_1, \dots, x_n]$ generated by the $u_i$'s, we are interested infinding the unique polynomial $f_{\rm new}$ in $\mathbb{K}[e_1,\dots, e_n]$, where $e_1, \dots, e_n$ are new variables, such that $f_{\mathrm{new}}(u_1, \dots, u_n) = f(x_1, \dots, x_n)$. We provide an algorithm and analyze its arithmetic complexity to compute $f_{\mathrm{new}}$ knowing $f$ and $(u_1, \dots, u_n)$.
