Classifying real polynomial pencils

Details Classifying real polynomial pencils Classifying real polynomial pencils

2004-04-11

arxiv.org/abs/math/0404215v1

Int. Math. Res. Not. 69 (2004), 3689--3708

Mathematics

Algebraic Geometry

15 pages, 7 figures, LaTeX2e

Scientific paper

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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