论文标题
关于产品分解
On product decomposition
论文作者
论文摘要
给定有限集合$ w $ in $ \ bar {k}^n $其中$ \ bar {k} $是一个field $ k $的代数闭合，一个人希望确定是否可以将$ w $分解为$ \ prod_ {i = 1}^n v_i $ v_i $ v_i $ v_i \ subset \ subset \ subset \ bar { $ w \stackrelλ{\ to} \ prod_ {i = 1}^n v_i $其中$λ\在gl_n中（\ bar {k}）$。我们假设$ W $以$ W = z（\ Mathcal {f}）$表示，$ n $变量在$ k $的$ n $变量中的零集$ \ MATHCAL {f} $。我们研究这种产品分解的代数表征。对于分解为相同基数的组件集，我们获得了更强的表征，并表明在这种情况下的分解本质上是唯一的（符合坐标的排列和标量乘法）。我们研究了分解问题引起的计算问题。
Given a finite set $W$ in $\bar{k}^n$ where $\bar{k}$ is the algebraic closure of a field $k$ one would like to determine if $W$ can be decomposed as $\prod_{i=1}^n V_i$ where $V_i \subset \bar{k}$ under a linear transformation, that is, $W\stackrelλ{\to} \prod_{i=1}^n V_i$ where $λ\in Gl_n (\bar{k})$. We assume that $W$ is presented as $W=Z(\mathcal{F})$, the zero set of a polynomial system $\mathcal{F}$ in $n$ variables over $k$. We study algebraic characterization of such product decomposition. For decomposition into component sets of the same cardinality we obtain a stronger characterization and show that the decomposition in this case is essentially unique (up to permutation and scalar multiplication of coordinates). We investigate computational problems that arise from the decomposition problem.
