论文标题
平面多项式和哈密顿矢量场由它们的奇异点决定
Plane polynomials and Hamiltonian vector fields determined by their singular points
论文作者
论文摘要
令$σ（f）$为\ mathbb {k} [k} [x，y] $的多项式$ f \ in Plane $ \ Mathbb {k}^2 $中的关键点，其中$ \ mathbb {k {k} $ is $ \ \ \ m mathbb {r Mathbb {r} $ {r} $或$ \ mathbbbb {r mathbb {c} $ {c} $ {c} $ {c} $。我们的目标是通过向其关键点$σ（f）$发送polyenmials $ f $ $ d $来研究关键点图$ \ mathfrak {s} _d $。非常粗略地说，当任何其他$ g $共享$ f $的关键点满足$ f =λg$时，都可以确定多项式$ f $；这两个都是多项式，最多是$ d $，$λ\ in \ mathbb {k}^*$。为了描述$ d $本质上确定的多项式，提供了所需数量的隔离临界点$δ（d）$的计算。出现$δ（d）$的值的二分法；根据一定的平等，基本确定的多项式的空间是开放或封闭的Zariski集。我们计算地图$ \ mathfrak {s} _3 $，描述在什么条件下，四个点的配置导致三个学位三个基本确定的多项式。此外，我们描述了支撑三个非基本确定多项式的明确配置。在本质上确定的三级多项式的商空间，直到仿射组$ \ hbox {aff}（\ mathbb {k}^2）$的动作确定$ \ mathbb {k} $上的单数表面。
Let $Σ(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials $f$ of degree $d$ to their critical points $Σ(f)$ . Very roughly speaking, a polynomial $f$ is essentially determined when any other $g$ sharing the critical points of $f$ satisfies that $f= λg$; here both are polynomials of at most degree $d$, $λ\in \mathbb{K}^*$. In order to describe the degree $d$ essentially determined polynomials, a computation of the required number of isolated critical points $δ(d)$ is provided. A dichotomy appears for the values of $δ(d)$; depending on a certain parity the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $\mathfrak{S}_3$, describing under what conditions a configuration of four points leads to a degree three essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree three non essential determined polynomials. The quotient space of essentially determined polynomials of degree three up to the action of the affine group $\hbox{Aff}(\mathbb{K}^2)$ determines a singular surface over $\mathbb{K}$.
