The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability
By Julius Borcea, Petter Brand'en. 2008
171 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/5b538cf255f8ec51d32ce78cd75ce5e409847079
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In Applied Mathematics
In Discrete Mathematics and Combinatorics
Citations in the corpus
- Negative Dependence in Sampling (2012, unknown): “1 of [1], the claim is equivalent to stability of the polynomial (in 2N variables)”
- Multivariate stable polynomials: theory and applications (2009, background): “by f˝g, when g+if2S[x]. (As will be seen, this implies that f;g2S R[x].) Thus, the multivariate analogue of Hermite-Biehler is a denition, not a theorem. Proposition 2.7 (Lemma 1.8 and Remark 1.3 of [5]). Let f;g2C[x]. (a) If f;g2R[x] then f˝gif and only if g+ yf2S R[x;y]. (b) If 0 6f2S[x] then g+ y”
- Unimodality, log-concavity, real-rootedness and beyond (2015, background): “d y = (y2;:::;y n). To prove that A n(x;y) is stable for all nit remains to prove that the operators of the form P n j=1 @=@x j preserve stability. Stability preservers were recently characterized in [9]. The following theorem is the algebraic characterization. For 2Nn, let C [x1;:::;x n] be the lin”