The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability
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 P\'olya-Schur on univariate polynomials with such properties.
Keywords
Cite
@article{arxiv.0809.0401,
title = {The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability},
author = {Julius Borcea and Petter Brändén},
journal= {arXiv preprint arXiv:0809.0401},
year = {2012}
}
Comments
Final version, to appear in Inventiones Mathematicae; 27 pages, no figures, LaTeX2e