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

Abstract

We develop tools to study, understand and 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 the matrices of the relaxation. We develop tools to study, understand and 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 the matrices of the relaxation. 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 Eulerian polynomials. As the univariate Eulerian polynomials present symmetries that helped us to understand better the structure of their roots, we also study generalizations of these notions of symmetry to multivariate Eulerian polynomials. These generalizations of symmetry shed new light on the combinatorial object these polynomials encode. Finally, the combination of all these techniques coming from real algebraic geometry, the theory of stability preservers, the numerical methods for root estimations and the study of symmetries of polynomials encoding combinatorial objects that we use in this thesis along the path suggested by Eulerian polynomials suggests to us that there is much more happening under the surface. This additional knowledge would affect and increase our understanding of many more families of polynomials emerging as generating polynomials associated to combinatorial objects, this is, in the same way as Eulerian polynomials emerge. Thus, the insights provided here by the study of Eulerian polynomials allow us to propose a further reaching mathematical program aimed at studying and delving deeper into the relations, connections and phenomena first devised and exemplified by the work presented here on Eulerian polynomials. We name this program the Mindelsee Program and find that it encourages connecting paths that could open bridges between many different areas of mathematics promoting thus the study of new families of (mainly real zero) polynomials and of new techniques aimed at improving our abilities to perform the task of understanding better the overall structure of their roots in depth.