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
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.
stable polynomial, zeros of a complex polynomial, total nonnegative Grassmannian, totally nonnegative matrix
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