Computational Methods and Function Theory 1 (2001), No. 1, 1--25
Copyright Heldermann Verlag 2001
Alexandre Eremenko
eremenko@math.purdue.edu, Purdue University, West Lafayette, Indiana 47907, U.S.A.
Andrei Gabrielov
agabriel@math.purdue.edu, Purdue University, West Lafayette, Indiana 47907, U.S.A.
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 $\pm u((m+1)/2)$ 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.
Keywords: Rational functions, Grassmann varieties, enumerative geometry, control of linear systems.
MSC 2000: 26C15, 14N10, 14N15, 14M15, 93C05, 93C35.
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