Classifying real polynomial pencils

2004/04/11 by Julius Borcea, Boris Shapiro · 1 citation
Mathematics · #math.AG #math.CA #msc:58K05 #msc:12D10 #msc:14P05 #msc:26C10 #msc:30C15

paper · pdf

published as Int. Math. Res. Not. 69 (2004), 3689--3708 · 15 pages, 7 figures, LaTeX2e

arxiv created 2004/04/11 · arxiv updated 2009/12/01

Abstract

Let \bPⁿ be the space of all homogeneous polynomials of degree n in two variables with real coefficients. The standard discriminant \D_n+1⊂ \bPⁿ is Whitney stratified according to the number and the multiplicities of multiple real zeros. A real polynomial pencil, that is, a line L⊂ \bPⁿ is called generic if it intersects \D_n+1 transversally. Nongeneric pencils form the Grassmann discriminant \D_2,n+1⊂ \gtn, where \gtn is the Grassmannian of lines in \bPⁿ. We enumerate the connected components of the set \widetilde \gtn=\gtn∖ \D_2,n+1 of all generic lines in \bPⁿ and relate this topic to the Hawaii conjecture and the classical theorems of Obreschkoff and Hermite-Biehler.

Cited by

• Complete Resolution of B.Shapiro's Conjecture 12

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