Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alejandro González Nevado

TL;DR

We study global bounds for the extreme roots of multivariate real zero polynomials using a spectrahedral relaxation built from a degree $3$ truncation, yielding a tractable representation whose size depends only on the number of variables. The approach analyzes two main directions to enrich information in the relaxation: extending the variable set and extending the moment-matrix mold, with careful attention to PSD-ness and invariances. Focusing on multivariate Eulerian polynomials, injecting univariate Eulerian diagonally into the multivariate framework yields exponential improvements in the bounds, surpassing existing results and aligning closely with the true roots. The work integrates real algebraic geometry, stability preservers, and interlacing techniques, and proposes the Mindelsee program as a broader research roadmap for connecting combinatorial polynomials to spectral representations and root structure across families. Together, these contributions provide asymptotically accurate root bounds for Eulerian and related real-zero polynomials and open pathways for broader applications in convex algebraic geometry and combinatorics.

Abstract

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.

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

TL;DR

We study global bounds for the extreme roots of multivariate real zero polynomials using a spectrahedral relaxation built from a degree truncation, yielding a tractable representation whose size depends only on the number of variables. The approach analyzes two main directions to enrich information in the relaxation: extending the variable set and extending the moment-matrix mold, with careful attention to PSD-ness and invariances. Focusing on multivariate Eulerian polynomials, injecting univariate Eulerian diagonally into the multivariate framework yields exponential improvements in the bounds, surpassing existing results and aligning closely with the true roots. The work integrates real algebraic geometry, stability preservers, and interlacing techniques, and proposes the Mindelsee program as a broader research roadmap for connecting combinatorial polynomials to spectral representations and root structure across families. Together, these contributions provide asymptotically accurate root bounds for Eulerian and related real-zero polynomials and open pathways for broader applications in convex algebraic geometry and combinatorics.

Abstract

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 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.

Paper Structure

This paper contains 69 sections, 109 theorems, 308 equations, 1 figure.

Table of Contents

  1. Preamble to Understand the Architecture of this Dissertation
  2. A mathological foreword
  3. Prolegomena to any attempt to read this work
  4. Introduction to the intersection of subjects meeting here
  5. Mathematical prerequisites necessary before continuing
  6. Structure of the contents
  7. Main results and a look beyond
  8. On the Definition of the Relaxation and its Limitations
  9. Relaxation of Rigidly Convex Sets
  10. Main definitions for real zero polynomials
  11. Motivation for the necessity and usability of a relaxation
  12. Extending the logarithm and the exponential
  13. Defining the relaxation
  14. Key properties of the relaxation
  15. Variations of the relaxation
  16. ...and 54 more sections

Key Result

Proposition 6.1

Applying $L_{p,e}$ for any integer $e\geq d:=\deg(p)$ to all the entries of any MMM of the form given by Equation ext1 for any RZ polynomial $p\in\mathbb{R}_{\hbox{RZ}}[\mathbf{x}]$ produces a PSD matrix.

Figures (1)

  • Figure 1: Representation of the entries of the eigenvectors in the interval $[0,1]$ for the relaxation corresponding to the multivariate Eulerian polynomials of degrees 1 to 10.

Theorems & Definitions (315)

  • Remark 1.1: Fortifications of the abilities of the hammer
  • Remark 1.2: Using several hammers to pin one nail
  • Remark 1.3: Symmetries, combinations and beyond
  • Remark 3.1: RAG and optimization
  • Remark 3.2: Trace inequalities, non-commutative, interlacers and polynomial stability
  • Remark 3.3: Eulerian polynomials and statistics on permutations over ordered sets
  • Remark 3.4: Root bounds, asymptotic expansions and numerics of generalized eigenvalue problems
  • Remark 3.5: (Approximated) computational complexity and numerical stability
  • Remark 3.6: Numerics and distant bits of probability and analysis
  • Remark 3.7: Symmetries and patterns in polynomials
  • ...and 305 more