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

• 成果类型: Article
• 署名作者: Borcea, Julius; Branden, Petter
• 署名单位: Royal Institute of Technology; Stockholm University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0189-3
• 发表日期: 2009
• 页码: 541-569
• 关键词: 1st-order phase-transitions; partition-function zeros; POLYNOMIALS; SEQUENCES; roots

摘要

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 Polya-Schur on univariate polynomials with such properties.