论文标题
打结的表面作为消失的多项式
Knotted surfaces as vanishing sets of polynomials
论文作者
论文摘要
我们提出了一种算法，该算法将循环编织组的任何元素$ b $作为输入，并构造了多项式$ f：\ mathbb {r}^5 \ to \ to \ mathbb {r}^2 $，从而使$ f $的消失集的相交包含$ f $的消失集和4个单位包含$ b $ $ b $的交易。多项式可用于创建具有零差异和闭合流线的真实分析时间相关的向量字段，这些载流线被$ b $所规定的移动。我们还展示了如何在没有分支点的情况下以$ \ mathbb {c} \ times s^1 \ times s^1 $构建的表面辫子家族，以构建为消失的全态多项式$ f：\ mathbb {c}^3 \ to \ mathbb {c} $ oon $ \ mathbb { s^1 \ subset \ mathbb {c}^3 $。这两种结构都使我们可以在多项式的程度上给出上限。
We present an algorithm that takes as input any element $B$ of the loop braid group and constructs a polynomial $f:\mathbb{R}^5\to\mathbb{R}^2$ such that the intersection of the vanishing set of $f$ and the unit 4-sphere contains the closure of $B$. The polynomials can be used to create real analytic time-dependent vector fields with zero divergence and closed flow lines that move as prescribed by $B$. We also show how a family of surface braids in $\mathbb{C}\times S^1\times S^1$ without branch points can be constructed as the vanishing set of a holomorphic polynomial $f:\mathbb{C}^3\to\mathbb{C}$ on $\mathbb{C}\times S^1\times S^1\subset\mathbb{C}^3$. Both constructions allow us to give upper bounds on the degree of the polynomials.
