Current Developments in Mathematics

Volume 2011

Hyperbolicity and stable polynomials in combinatorics and probability



Robin Pemantle (University of Pennsylvania)


These lectures survey the theory of hyperbolic and stable polynomials, from their origins in the theory of linear PDEs to their present uses in combinatorics and probability theory.


amoeba; cone; dual cone; Gaarding-hyperbolicity; generating function; half-plane property; homogeneous polynomial; Laguerre-Pólya class; multiplier sequence; multi-affine; multivariate stability; negative dependence; negative association; Newton’s inequalities; Rayleigh property; real roots;semi-continuity; stochastic covering; stochastic domination; total positivity

2010 Mathematics Subject Classification

Primary 26C10, 62H20. Secondary 05A15, 30C15.

Published 15 March 2013