Stable multivariate W-Eulerian polynomials

We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type D. nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D. we indicate some methods of attack and pose some related open problems. (C) 2013 Elsevier Inc. reserved.