Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moment inequalities for Chow polynomials of matroids and bounds on Chern numbers

Ronnie Cheng, Wangyang Lin

Abstract

The Chow polynomial of a matroid is a fundamental invariant whose coefficients exhibit strong positivity properties, including $γ$-positivity. We interpret the normalized Chow coefficients as a probability distribution and establish new inequalities for its central moments. As consequences, we obtain bounds on the number of flags of flats and inequalities on the roots of the Chow polynomial. We further relate these moment inequalities to algebraic geometry via the Hirzebruch $χ_y$-genus. This yields new inequalities for matroidal Chern numbers. In particular, for any matroid of rank $d+1$, we prove that $c_1c_{d-1}\le c_d$, with equality if and only if $d=1$ or the simplification of the matroid is Boolean.

Moment inequalities for Chow polynomials of matroids and bounds on Chern numbers

Abstract

The Chow polynomial of a matroid is a fundamental invariant whose coefficients exhibit strong positivity properties, including -positivity. We interpret the normalized Chow coefficients as a probability distribution and establish new inequalities for its central moments. As consequences, we obtain bounds on the number of flags of flats and inequalities on the roots of the Chow polynomial. We further relate these moment inequalities to algebraic geometry via the Hirzebruch -genus. This yields new inequalities for matroidal Chern numbers. In particular, for any matroid of rank , we prove that , with equality if and only if or the simplification of the matroid is Boolean.
Paper Structure (20 sections, 48 theorems, 185 equations, 1 algorithm)

This paper contains 20 sections, 48 theorems, 185 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The wonderful variety
  4. Chow ring and K-ring of a matroid
  5. Tangent class of a matroid
  6. Valuativeness of flag-countings and Chern classes
  7. Chow polynomial and its semi-small decomposition
  8. Moment inequalities for Chow coefficients
  9. Properties of Eulerian numbers
  10. Bounding central moments via the normal distribution
  11. The naive bound and the numbers of flags of flats
  12. Connection to $\gamma$-coefficients of the Chow polynomial
  13. Constraints on roots and the real-rootedness conjecture
  14. Chern classes of matroids
  15. Interpretation as inequalities of Chern numbers
  16. ...and 5 more sections

Key Result

Theorem A

Let $\mathcal{X}_d$ denote the random variable associated with the Boolean matroid $\mathrm{U}_{d+1}=\mathrm{U}_{d+1,d+1}$. Suppose that a sequence of functions $\{f_0,\dots,f_t\}$ forms a CMFS such that for all $d\ge0$ and $0\le k\le t$, Then for any matroid $\mathrm{M}$ of rank $d+1$,

Theorems & Definitions (99)

  • Theorem A: Theorem \ref{['thm:generalineq']}
  • Corollary B: Corollary \ref{['cor:ineq_chow_coefficient']}
  • Theorem C: Theorem \ref{['thm:main-theorem']}
  • Lemma D: Lemma \ref{['lem:hirzeformula']}
  • Theorem E: Corollary \ref{['cor:ineq_of_chern_numbers']}
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3: EHL_Stella or DF_val
  • Definition 2.4
  • Proposition 2.5
  • ...and 89 more