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Inventiones mathematicae

, 177:541 | Cite as

The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability

  • Julius Borcea
  • Petter BrändénEmail author
Article

Abstract

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Pólya-Schur on univariate polynomials with such properties.

Mathematics Subject Classification (2000)

47B38 05A15 05C70 30C15 32A60 46E22 82B20 82B26 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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