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Explicit expressions for the gamma vector leading to connections to upper/lower bounds and structural properties

Soohyun Park

Abstract

We find an explicit formula for the gamma vector in terms of the input polynomial in a way that extends it to arbitrary polynomials. More specifically, we find explicit linear combination in terms of coefficients of the input polynomial (using Catalan numbers and binomial coefficients) and an expression involving the derivative of the input polynomial. The first expression suggests connections to common Coxeter group/noncrossing partition structures in existing gamma positivity examples. In the case where the input is the $h$-polynomial of a simplicial complex, this gives an interpretation of the gamma vector as a measure of differences in local and global contributions. We also apply them to connect signs/inequalities of (shifts of) the gamma vector to upper/lower bound conditions on coefficients of the input polynomial. Finally, we make use of the shape of the sums used to make these estimates and connections with intersection numbers to relate these properties of the gamma vector to algebraic structures (e.g. characteristic classes involved in existing log concavity and Schur positivity properties).

Explicit expressions for the gamma vector leading to connections to upper/lower bounds and structural properties

Abstract

We find an explicit formula for the gamma vector in terms of the input polynomial in a way that extends it to arbitrary polynomials. More specifically, we find explicit linear combination in terms of coefficients of the input polynomial (using Catalan numbers and binomial coefficients) and an expression involving the derivative of the input polynomial. The first expression suggests connections to common Coxeter group/noncrossing partition structures in existing gamma positivity examples. In the case where the input is the -polynomial of a simplicial complex, this gives an interpretation of the gamma vector as a measure of differences in local and global contributions. We also apply them to connect signs/inequalities of (shifts of) the gamma vector to upper/lower bound conditions on coefficients of the input polynomial. Finally, we make use of the shape of the sums used to make these estimates and connections with intersection numbers to relate these properties of the gamma vector to algebraic structures (e.g. characteristic classes involved in existing log concavity and Schur positivity properties).
Paper Structure (8 sections, 19 theorems, 79 equations)

This paper contains 8 sections, 19 theorems, 79 equations.

Table of Contents

  1. The gamma vector as a linear combination of the $h$-vector components and local-global differences
  2. An explicit formula involving Catalan numbers, binomial coefficients, and derivatives
  3. Local-global differences
  4. Inequalities and auxiliary $h$-vector representations for gamma vectors of polynomials
  5. Bounds involving translations
  6. Upper/lower bounds and general gamma vectors
  7. Gamma vectors and (modified) $h$-vector representations
  8. Intersection numbers and volume polynomials

Key Result

Theorem 1.1

Writing $n = \deg h$ for a reciprocal polynomial $h(t)$, we have that where $\widetilde{C}_k$ is the $k^{\text{th}}$ Catalan number if $k \ge 1$ and $\widetilde{C}_0 = 0$. Since $\deg \gamma \le \lfloor \frac{n}{2} \rfloor$ if $h$ is reciprocal, this sum is equal to $0$ for $m > \lfloor \frac{n}{2} \rfloor$ in that case.

Theorems & Definitions (42)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • Theorem 1.7
  • proof
  • ...and 32 more