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Total Nonnegativity and Stable Polynomials

Published online by Cambridge University Press:  20 November 2018

Kevin Purbhoo
Kevin Purbhoo*
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. W., Waterloo, ON N2L 3G1, e-mail : kpurbhoo@uwaterloo.ca
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Abstract

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We consider homogeneous multiaffine polynomials whose coefficients are the Plücker 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 Pólya-Schur theory of Borcea and Brändén.

Keywords

32A6014M1514P1015B48stable polynomialzeros of a complex polynomialtotal nonnegative Grassmanniantotally nonnegative matrix
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 4 , 01 December 2018 , pp. 836 - 847
Copyright
Copyright © Canadian Mathematical Society 2018

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