CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Author's Draft

Total nonnegativity and stable polynomials

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

© Canadian Mathematical Society, 2018 : https://cms.math.ca/