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Shellable slices of hyperbolic polynomials and the degree principle

Arne Lien, Robin Schabert

Abstract

We study a natural stratification of certain affine slices of univariate hyperbolic polynomials. We look into which posets of strata can be realized and show that the dual of the poset of strata is a shellable simplicial complex and in particular a combinatorial sphere. From this we obtain a g-theorem and an upper bound theorem on the number of strata. We use these results to design smaller test sets to improve upon Timofte's degree principle and give bounds on how much the degree principle can be improved.

Shellable slices of hyperbolic polynomials and the degree principle

Abstract

We study a natural stratification of certain affine slices of univariate hyperbolic polynomials. We look into which posets of strata can be realized and show that the dual of the poset of strata is a shellable simplicial complex and in particular a combinatorial sphere. From this we obtain a g-theorem and an upper bound theorem on the number of strata. We use these results to design smaller test sets to improve upon Timofte's degree principle and give bounds on how much the degree principle can be improved.
Paper Structure (14 sections, 36 theorems, 80 equations, 1 figure)

This paper contains 14 sections, 36 theorems, 80 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Simplicial complexes, shellings and spheres
  3. Symmetric polynomials and Vandermonde varieties
  4. Hyperbolic slices and posets
  5. Stratification of hyperbolic slices
  6. Escaping hyperbolic strata
  7. Shellability of the dual poset
  8. Shelling the dual
  9. UBT and g-theorem
  10. Improving Timofte's Degree principle
  11. General bounds on Vandermonde coverings
  12. Algorithmic improvements of Vandermonde coverings.
  13. Conclusion
  14. Proof of Theorem \ref{['2:thm:minmax']}

Key Result

Proposition 1.7

A shellable simplicial complex of dimension $m$, whose ridges are all contained in exactly two facets, is a combinatorial $m$-sphere.

Figures (1)

  • Figure 1:

Theorems & Definitions (104)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Proposition 1.7
  • Definition 1.8
  • Theorem 1.9: Fundamental Theorem of Symmetric Polynomials
  • proof
  • ...and 94 more