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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Julius Borcea

  2. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Petter Brändén

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  1. Julius Borcea
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  2. Petter Brändén
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Correspondence to Petter Brändén.

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The first author is supported by the Royal Swedish Academy of Sciences. The second author is supported by the Göran Gustafsson Foundation.

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Borcea, J., Brändén, P. The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability. Invent. math. 177, 541–569 (2009). https://doi.org/10.1007/s00222-009-0189-3

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