A Method for Establishing Asymptotically Accurate Bounds for Extremal Roots of Eulerian Polynomials Using Polynomial Stability Preservers

We develop the tools to bound extreme roots of multivariate real zero polynomials globally. This is done through the use of a relaxation that approximates their rigidly convex sets. This relaxation can easily be constructed using the degree $3$ truncation of the polynomial and it produces in this way a spectrahedron whose computation is relatively easy and whose size is relatively small and depending solely on the number of variables of the polynomial. As we know that, in order to be able to produce in general spectrahedral representations of rigidly convex sets it is necessary to build matrices of very big size, we try, analyze and experiment with several constructions that could increase the size of these matrices. These constructions are based principally in two main approaches: adding information about higher degree monomials or non-trivially increasing the number of variables of the original polynomial. We explore these two construction first in a general setting and see that it is necessary to particularize to certain families of polynomials in order to make them work. In particular, we are able to prove that increasing the number of variables improves the behavior of the relaxation along the diagonal in the case of Eulerian polynomials. We see that applying the relaxation to multivariate Eulerian polynomials and then looking at the univariate polynomials injected in their diagonals produces an exponential asymptotic improvement in the bounds provided. We compare these bounds with other bounds that have appeared previously in the literature and refine these previous bounds in order to study how close do the bounds provided by the relaxation are to the actual roots of the univariate Eulerian polynomials.