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arXiv · 1611.07548

Total nonnegativity and stable polynomials

Kevin Purbhoo
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Abstract

We consider homogeneous multiaffine polynomials whose coefficients are the Pl\"ucker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized P\'olya-Schur theory of Borcea and Br\"and\'en.

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Kevin Purbhoo. 2016-11-22. Total nonnegativity and stable polynomials. https://doi.org/10.4153/cmb-2018-006-7

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