论文标题
单变量的辫子单构单元
Braid monodromy of univariate fewnomials
论文作者
论文摘要
令$ \ mathcal {c} _d \ subset \ mathbb {c}^{d+1} $是$ d $的非单词，单变量多项式的空间。 viète映射$ \ mathscr {v}：\ mathcal {c} _d \ rightArrow sym_d（\ mathbb {c}）$将多项式发送到其无序的根源集。这是一个经典的事实，即在基本组的级别上诱导的地图$ \ mathscr {v} _*$实现了$π_1（\ Mathcal {c} _d _d）$与Artin Braid $ b_d $之间的同构。对于几个nomials，或等同于交叉点$ \ nathcal {c} $ $ \ Mathcal {c} _d $，带有$ \ Mathbb {c}^{d+1} $ in Mathbb {c}^{d+1} $中的坐标超平面的图像B_D $一般不知道。在本文中，我们表明地图$ \ mathscr {v} _*$是围绕的，只要相应的多项式的支持跨越$ \ mathbb {z} $作为仿射晶格。如果支持跨越了索引$ b $的严格sublattice，我们表明$ \ mathscr {v} _*$的图像是$ \ mathbb {z}/b \ mathbb {z} $带有$ b_ {d/b} $的预期花圈产品。从这些结果中，我们根据一组共同的参数来得出辫子单变量的计算，以计算单变量多项式。
Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Viète map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $\mathscr{V}_*$ at the level of fundamental groups realises an isomorphism between $π_1(\mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{C}$ of $\mathcal{C}_d$ with a collection of coordinate hyperplanes in $\mathbb{C}^{d+1}$, the image of the map $\mathscr{V} _* : π_1(\mathcal{C}) \rightarrow B_d$ is not known in general. In the present paper, we show that the map $\mathscr{V} _*$ is surjective provided that the support of the corresponding polynomials spans $\mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of $\mathscr{V} _*$ is the expected wreath product of $\mathbb{Z}/b\mathbb{Z}$ with $B_{d/b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.
