Classifying real polynomial pencils

TL;DR

This paper classifies generic real polynomial pencils by analyzing their intersections with the discriminant, connecting the results to classical theorems and conjectures in real algebraic geometry.

Contribution

It provides a enumeration of connected components of generic polynomial pencils and relates these to classical theorems and the Hawaii conjecture.

Findings

Connected components of generic pencils are enumerated.

Relation established between polynomial pencils and classical theorems.

Connections made to the Hawaii conjecture and Obreschkoff's and Hermite-Biehler's theorems.

Abstract

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.