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# Total Nonnegativity and Stable Polynomials

Published:2018-07-06
Printed: Dec 2018
• Kevin Purbhoo,
Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. W., Waterloo, ON N2L 3G1
 Format: LaTeX MathJax PDF

## Abstract

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: stable polynomial, zeros of a complex polynomial, total nonnegative Grassmannian, totally nonnegative matrix
 MSC Classifications: 32A60 - Zero sets of holomorphic functions 14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14P10 - Semialgebraic sets and related spaces 15B48 - Positive matrices and their generalizations; cones of matrices

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