Multivariate stable polynomials: theory and applications

By David G. Wagner. 2009

182 citations recorded by the source corpus.

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 Br\"and\'en, 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/05b89a39ee536a4fafb488edcccab1f3523a220b

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Citations in the corpus

Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees (2013, background): “Then the ( univariate ) polynomial Proof .”
Logarithmic concavity of Schur and related polynomials (2019, background): “ 2qx 2 x 1x 2 x 2 2 : The quadratic form has eigenvalues 3 2 and 1 2 , and hence s is not Lorentzian. A polynomial fpx 1;:::;x mqis stable if fhas no zeros in the product of mopen upper half planes [Wag11]. Homogeneous stable polynomials with nonnegative coefﬁcients are motivating examples of Lorent”
Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem (2013, methodology): “We will also use the fact that real stability is preserved under setting variables to real numbers (see, for instance, [48, Lemma 2.4(d)]).”
Unimodality, log-concavity, real-rootedness and beyond (2015, methodology): “s Gurvits proof of a vast generalization of the Van der Waerden conjecture, [54]. A recent application is the spectacular solution to the Kadison{Singer problem by Marcus et al. [67]. See the surveys [76,104] for further applications of stable polynomials. The notion of HPP and WHPP matroids were in”
Symmetric Decompositions and Real-Rootedness (2018, methodology): “To do so, we require a few tools from the theory of (real) stable polynomials ; a multivariate generalization of real-rooted polynomials which was developed in [8, 9] and surveyed in [41].”