The Wronski map and Grassmannians of Real Codimension 2 Subspaces

Abstract

We study the map which sends a pair of real polynomials (f 0 , f 1 ) into their Wronski determinant W (f 0 , f 1 ). This map is closely related to a linear projection from a Grassmannian G R (m, m + 2) to the real projective space RP 2m . We show that the degree of this projection is ±u((m+1)/2) where u is the m-th Catalan number. One application of this result is to the problem of describing all real rational functions of given degree m + 1 with prescribed 2m critical points. A related question of control theory is also discussed.

Rational functions and real Schubert calculus

“…Enumerate subspaces X ⊂ P d of codimension 2 which intersect each W j non-transversally, that is, (2) dim X ∩ W j ≥ a j − 1 for every j ∈ [1, q].…”

Degrees of Real Wronski Maps

“…The lower bound in Corollary 2 with p = 2 is best possible, as the following example given in [4] shows:…”

“…For the case p = 2, the results of this paper were obtained in [4], with a different method based on [3]. We thank S. Fomin, Ch.…”

An Elementary Proof of the B. and M. Shapiro Conjecture for Rational Functions

“…If C = C(g) then deg C = deg g. A net of degree d has 2d − 2 edges disjoint from the real axis. Using the Uniformization theorem, we proved in [1,2] that each net comes from a rational function, and the critical points of this rational function can be arbitrarily prescribed, but we do not use this result here, and in fact it will be deduced from our theorems 1 and 2 in the end of Section 1.…”