The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability

Details The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability

2008-09-02

arxiv.org/abs/0809.0401v3

Invent. Math. 177 (2009), no. 3, 541-569

Mathematics

Complex Variables

Final version, to appear in Inventiones Mathematicae; 27 pages, no figures, LaTeX2e

Scientific paper

10.1007/s00222-009-0189-3

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 P\'olya-Schur on univariate polynomials with such properties.

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