Geodesic
Total Nonnegativity and Stable Polynomials
Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 836-847

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DOI

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.
DOI : 10.4153/CMB-2018-006-7
Mots-clés : 32A60, 14M15, 14P10, 15B48, stable polynomial, zeros of a complex polynomial, total nonnegative Grassmannian, totally nonnegative matrix 
Purbhoo, Kevin. Total Nonnegativity and Stable Polynomials. Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 836-847. doi: 10.4153/CMB-2018-006-7
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     author = {Purbhoo, Kevin},
     title = {Total {Nonnegativity} and {Stable} {Polynomials}},
     journal = {Canadian mathematical bulletin},
     pages = {836--847},
     year = {2018},
     volume = {61},
     number = {4},
     doi = {10.4153/CMB-2018-006-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2018-006-7/}
}
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