2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174150We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory. This is an announcement with some of the main results in arXiv:0809.0401 and arXiv:0809.3087.To appear in Letters in Mathematical Physics; 8 pages, no figures, LaTeX2eComplex VariablesStatistical MechanicsMathematical PhysicsCombinatorics47B38 (Primary);05A15, 05C70, 30C15, 32A60, 46E22, 82B20, 82B26 (Secondary)text