分类实多项式束
Classifying real polynomial pencils
摘要：
设$\bP^n$是二元实系数齐次多项式$n$的空间。标准判别式$\D_{n+1}\subset \bP^n$是根据多个实零点的数量和重数进行Whitney分层的。一个实多项式束，即一条线$L\subset \bP^n$，如果它与$\D_{n+1}$横向相交，则称为泛型。非属笔形成格拉斯曼判别式$\D_{2,n+1}\subset \gtn$，其中$\gtn$是$\bP^n$中的线的格拉斯曼数。我们列举了$\bP^n$中所有一般线的集合$\widetilde \gtn=\gtn\setminus \D_{2,n+1}$的连通分支，并将此主题与夏威夷猜想以及Obreschkoff和Hermite-Biehler的经典定理联系起来。
英文摘要：
Let $\bP^n$ be the space of all homogeneous polynomials of degree $n$ in two variables with real coefficients. The standard discriminant $\D_{n+1}\subset \bP^n$ is Whitney stratified according to the number and the multiplicities of multiple real zeros. A real polynomial pencil, that is, a line $L\subset \bP^n$ is called generic if it intersects $\D_{n+1}$ transversally. Nongeneric pencils form the Grassmann discriminant $\D_{2,n+1}\subset \gtn$, where $\gtn$ is the Grassmannian of lines in $\bP^n$. We enumerate the connected components of the set $\widetilde \gtn=\gtn\setminus \D_{2,n+1}$ of all generic lines in $\bP^n$ and relate this topic to the Hawaii conjecture and the classical theorems of Obreschkoff and Hermite-Biehler.
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