论文标题
计算由高核心体不变多项式定义的代数系统的关键点
Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials
论文作者
论文摘要
令$ \ mathbb {k} $为特征零的字段，$ \ mathbb {k} [x_1，\ dots，x_n] $相应的多变量多项式环。给定$ s $多项式的顺序$ \ mathbf {f} =（f_1，\ dots，f_s）$和一个多项式$ ϕ $，全部in $ \ mathbb {k} [x_1，\ dots，\ dots，x_n] $带有$ s <n $，我们考虑$ s <n $，我们考虑$ s $ n $ w（f \ west）n n ot thing（c） $ \ mathbf {f} $ nishes和$ \ mathbf {f}的jacobian矩阵，相对于$ x_1，\ dots，x_n $，ϕ $没有完整的等级。这个问题在许多应用领域都起着至关重要的作用。 在本文中，我们关注的是，在签名的对称组$ b_n $的作用下，多项式都是不变的。我们介绍了一个称为{\ em hyperctahedral表示}的概念，以描述$ b_n $ invariant集。我们根据$ b_n $的轨道研究输入多项式对分配$ w（ϕ，\ mathbf {f}）$的不变性属性，然后设计算法的算法，其输出为{thypoctahedral表示} $ w（ϕ，\ mathbf {f}）$。我们算法的运行时间在输出所描述的积分总数中是多项式。
Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $ϕ$, all in $\mathbb{K}[x_1, \dots, x_n]$ with $s<n$, we consider the problem of computing the set $W(ϕ, \mathbf{f})$ of points at which $\mathbf{f}$ vanishes and the Jacobian matrix of $\mathbf{f}, ϕ$ with respect to $x_1, \dots, x_n$ does not have full rank. This problem plays an essential role in many application areas. In this paper we focus on a case where the polynomials are all invariant under the action of the signed symmetric group $B_n$. We introduce a notion called {\em hyperoctahedral representation} to describe $B_n$-invariant sets. We study the invariance properties of the input polynomials to split $W(ϕ, \mathbf{f})$ according to the orbits of $B_n$ and then design an algorithm whose output is a {hyperoctahedral representation} of $W(ϕ, \mathbf{f})$. The runtime of our algorithm is polynomial in the total number of points described by the output.
