Real Stability Testing
  We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials preserves real-rootedness. The proof exploits properties of hyperbolic polynomials to reduce real stability testing to testing nonnegativity of a finite number of polynomials on an interval. 
Prasad Raghavendra, Nick Ryder, Nikhil Srivastava
cs.DS