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

TL;DR

This paper derives explicit formulas for the gamma vector of polynomials, revealing connections to combinatorial structures, bounds, and algebraic properties, with applications to simplicial complexes and Coxeter groups.

Contribution

It provides a new explicit linear expression for the gamma vector applicable to any polynomial, linking it to combinatorial, geometric, and algebraic structures.

Findings

Explicit gamma vector formulas involving Catalan and binomial coefficients

Connections between gamma vector signs and coefficient bounds

Relations of gamma vector properties to algebraic structures like characteristic classes

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).