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Multivariate stable polynomials: theory and applications. (English) Zbl 1207.32006
Summary: Univariate polynomials with only real roots, while special, do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of this paper surveys some of the main results of this theory of multivariate stable polynomials–the most central of these results is the characterization of linear transformations preserving stability of polynomials. The second part presents various applications of this theory in complex analysis, matrix theory, probability and statistical mechanics, and combinatorics.

MSC:
32A60 Zero sets of holomorphic functions of several complex variables
05A20 Combinatorial inequalities
05B35 Combinatorial aspects of matroids and geometric lattices
15A45 Miscellaneous inequalities involving matrices
15B48 Positive matrices and their generalizations; cones of matrices
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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