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The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability

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arxiv 0809.0401 v3 pith:P3TYZLB7 submitted 2008-09-02 math.CV cond-mat.stat-mechmath-phmath.COmath.MP

classification math.CVcond-mat.stat-mechmath-phmath.COmath.MP
keywords linearoperatorspreservingnon-vanishingolya-schurpolynomialsprogramproperties
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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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  1. On Factorizing Aggregate Counting Distributions into Independent Latent Processes

    math-ph 2026-07 accept novelty 7.0 of 10

    Every aggregate counting distribution admits a poset of positive factorizations into independent latent processes, and the maximum latent entropy (factorization entropy) is attained at maximal atomizations.

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