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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Multivariate stable polynomials: theory and applications
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by David G. Wagner PDF
Bull. Amer. Math. Soc. 48 (2011), 53-84 Request permission

Abstract:

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.
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Additional Information
  • David G. Wagner
  • Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
  • Email: dgwagner@math.uwaterloo.ca
  • Received by editor(s): May 17, 2010
  • Received by editor(s) in revised form: June 12, 2010
  • Published electronically: October 4, 2010
  • Additional Notes: Research supported by NSERC Discovery Grant OGP0105392.

  • Dedicated: In memoriam of Julius Borcea
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 48 (2011), 53-84
  • MSC (2010): Primary 32A60; Secondary 05A20, 05B35, 15A45, 15A48, 60G55, 60K35
  • DOI: https://doi.org/10.1090/S0273-0979-2010-01321-5
  • MathSciNet review: 2738906