Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gamma vectors as inverted Chebyshev expansions, type A to B transformations, and connections to algebraic structures

Soohyun Park

TL;DR

The paper develops an inverted Chebyshev expansion framework for gamma vectors of reciprocal/palindromic polynomials, yielding a concrete real-rootedness criterion and establishing connections to Chebyshev varieties as sparse-torus analogues. It fuses algebraic and combinatorial methods by relating gamma vectors to $f$-vectors via Tchebyshev subdivisions, and shows how type $A$ to type $B$ Coxeter-complex transformations arise naturally from these expansions. A representation-theoretic viewpoint ties the gamma construction to $\mathfrak{sl}_2(\mathbb{C})$-representations and ce-indices, while a descent-statistics perspective links to quasisymmetric-function theory and topological invariants of posets and cross polytopes. Overall, the work provides a unified bridge between analytic properties of reciprocal polynomials, geometric-combinatorial subdivision procedures, and algebraic combinatorics, with implications for sparse polynomial root finding via Chebyshev varieties.

Abstract

Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear combination of Chebyshev polynomials in terms of real-rootedness of that of the reciprocal polynomial built out of an inverted scaled tuple of the coefficients with one fixed and the rest divided by 2. It can be taken as a counterpart for arbitrary dimensions of a recent result of Bel-Afia--Meroni--Telen on hyperbolicity of Chebyshev curves with respect to the origin. In general, Chebyshev varieties serve as a counterpart of toric varieties in sparse polynomial root finding. Apart from this, the inverted Chebyshev expansion also yields connections between intrinsic properties of the gamma vector construction and the geometric combinatorics of simplicial complexes and posets. We find this by applying work of Hetyei on Tchebyshev subdivisions and Tchebyshev posets. In particular, we find that the gamma vector transformation is closely related to $f$-vectors of simplicial complexes resulting from successive edge subdivisions that transform the type A Coxeter complex to the type B Coxeter complex. Lifting to this to a modification of $cd$-indices, we show that the gamma vector inverted Chebyshev polynomial expansion lifts to a sum of (subdivisions of) cross polytopes which can be computed using (topological) descent statistics. While there are many examples where gamma positivity involving descent statistics, it is interesting to note we only assume the input polynomial is reciprocal/palindromic. Finally, Chebyshev polynomials of the second kind from derivatives give connections to Hopf algebras and quasisymmetric functions along with Lefschetz-type maps induced by $\mathfrak{sl}_2(\mathbb{C})$-representations.

Gamma vectors as inverted Chebyshev expansions, type A to B transformations, and connections to algebraic structures

TL;DR

The paper develops an inverted Chebyshev expansion framework for gamma vectors of reciprocal/palindromic polynomials, yielding a concrete real-rootedness criterion and establishing connections to Chebyshev varieties as sparse-torus analogues. It fuses algebraic and combinatorial methods by relating gamma vectors to -vectors via Tchebyshev subdivisions, and shows how type to type Coxeter-complex transformations arise naturally from these expansions. A representation-theoretic viewpoint ties the gamma construction to -representations and ce-indices, while a descent-statistics perspective links to quasisymmetric-function theory and topological invariants of posets and cross polytopes. Overall, the work provides a unified bridge between analytic properties of reciprocal polynomials, geometric-combinatorial subdivision procedures, and algebraic combinatorics, with implications for sparse polynomial root finding via Chebyshev varieties.

Abstract

Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear combination of Chebyshev polynomials in terms of real-rootedness of that of the reciprocal polynomial built out of an inverted scaled tuple of the coefficients with one fixed and the rest divided by 2. It can be taken as a counterpart for arbitrary dimensions of a recent result of Bel-Afia--Meroni--Telen on hyperbolicity of Chebyshev curves with respect to the origin. In general, Chebyshev varieties serve as a counterpart of toric varieties in sparse polynomial root finding. Apart from this, the inverted Chebyshev expansion also yields connections between intrinsic properties of the gamma vector construction and the geometric combinatorics of simplicial complexes and posets. We find this by applying work of Hetyei on Tchebyshev subdivisions and Tchebyshev posets. In particular, we find that the gamma vector transformation is closely related to -vectors of simplicial complexes resulting from successive edge subdivisions that transform the type A Coxeter complex to the type B Coxeter complex. Lifting to this to a modification of -indices, we show that the gamma vector inverted Chebyshev polynomial expansion lifts to a sum of (subdivisions of) cross polytopes which can be computed using (topological) descent statistics. While there are many examples where gamma positivity involving descent statistics, it is interesting to note we only assume the input polynomial is reciprocal/palindromic. Finally, Chebyshev polynomials of the second kind from derivatives give connections to Hopf algebras and quasisymmetric functions along with Lefschetz-type maps induced by -representations.
Paper Structure (6 sections, 12 theorems, 27 equations)

This paper contains 6 sections, 12 theorems, 27 equations.

Table of Contents

  1. Inverted Chebyshev expansions
  2. Compatibility between Chebyshev polynomials, type $A$ to $B$ subdivisions, and other linear transformations $\mathbb{R}[x] \longrightarrow \mathbb{R}[x]$
  3. Type A to type B subdivisions
  4. Compatibility of Chebyshev polynomials with linear transformations of polynomial spaces and $\mathfrak{sl}_2(\mathbb{C})$-representations
  5. Intrinsic descent statistics and connections to quasisymmetric functions
  6. Decompositions into (subdivisions) of cross polytopes, (topological) descents, and Chebyshev polynomials of the second kind

Key Result

Proposition 1.2

Suppose that $d$ is even. For any $0 \le k \le \frac{d}{2}$, we have that for some numbers $\widetilde{\gamma}_i$ indexed over a finite set of nonnegative integers $I$ (depending on $\ell$) and they can be computed recursively. In particular, let $P_r = P_r(z)$ be the reciprocal polynomial $\sum_{i = 0}^{\frac{d}{2}} \widetilde{\gamma}_i^r w^{ \frac{d}{2} - i }$ of the mo

Theorems & Definitions (27)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Lemma 1.4
  • proof
  • Proposition 1.5
  • Remark 1.6
  • Theorem 1.7
  • ...and 17 more