Degree of the real Wronski map

Andrei Gabrielov

Purdue University

May 7,
refreshments at 3:45pm


The Wronski map associates to a $p$-tuple $Q=(q_1(x),\dots,q_p(x))$ of polynomials of degree at most $m+p-1$ their Wronski determinant $W(Q)$. If the polynomials are linearly independent, they define a $p$-dimensional subspace in the space of all polynomials, i.e., a point in the Grassmannian $G(p,m+p)$. Basic properties of the Wronskian imply that the Wronski map can be considered as a map from $G(p,m+p)$ to the projective space ${\bf P}^{mp}$. It respects the Schubert cell decomposition of $G(p,m+p)$, sending each cell $S$ to a projective subspace of the same dimension as $S$. The map is finite, and one can define its degree.

In the complex case, the degree of the Wronski map equals the number of standard tableaux for the Young diagram of a Schubert cell. In the real case, standard tableax should be counted with the signs depending on the number of inversions. For a rectangular Young diagram of the Grassmannian $G(p,m+p)$, the real degree is zero when $m+p$ is even, and equals the number of standard shifted Young tableaux for an appropriately defined shifted Young diagram when $m+p$ is odd. For $p=2$, the complex degree for the Grassmannian is $m$-th Catalan number, and the real degree is $\frac{m-1}{2}$-th Catalan number for odd $m$ and zero otherwise.

For a general Schubert cell, a closed form for the real degree is unknown. In particular, a question when this degree is not zero remains open. The answer to this question is important in applications to real Schubert calculus and to the pole placement problem in control theory.

Joint work with Alex Eremenko.

P.S. Most of the talk is linear algebra and manipulations with polynomials, and should be understandable by students.

Speaker's Contact Info: agabriel(at-sign)

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