Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets

Abstract: Stable multivariate Eulerian polynomials were introduced by Br\"and\'en. Particularizing some variables, it is possible to extract real zero multivariate Eulerian polynomials from them. These real zero multivariate Eulerian polynomials can be fed into constructions of spectrahedral relaxations providing therefore approximations to the (Eulerian) rigidly convex sets defined by these polynomials. The accuracy of these approximations is measured through the behaviour in the diagonal, where the usual univariate Eulerian polynomials sit. In particular, in this sense, the accuracy of the global spectrahedral approximation produced by the spectrahedral relaxation can be measured in terms of bounds for the extreme roots of univariate Eulerian polynomials. The bounds thus obtained beat the previous bounds found in the literature. However, the bound explicitly studied and obtained before beat the previously known bounds by a quantity going to $0$ when $n$ goes to infinity. Here we use numerical experiments to construct a sequence of vectors providing a (linearized) bound whose difference with the previous known bounds is a growing exponential function (going therefore fast to infinity when $n$ grows). This allows us to establish a better (diagonal) measure of accuracy for the spectrahedral relaxation of the Eulerian rigidly convex sets. In particular, we will achieve this by linearizing through the sequence of vectors ${(y,(-2^{m-i}){i=3}^{m},(0,\frac{1}{2}),(1){i=1}^{m})\in\mathbb{R}^{n+1}}_{n=1}^{\infty}$ for even $n=2m$.