On the real-rootedness of the Eulerian transformation

TL;DR

This paper resolves Brändén–Jochemko's conjecture by proving that the Eulerian transformation ${\mathcal A}^\circ$ preserves real-rootedness for all polynomials in ${\mathcal P}_n[x]$, establishing interlacing with $A_n(x)$ and $\widetilde{A}_n(x)$ and revealing a real-rooted, gamma-positive symmetric decomposition. It then embeds these phenomena into a broad combinatorial-geometric framework via uniform triangulations and barycentric subdivision, deriving gamma-positivity and unimodality results for generalized transformations ${\mathcal H}^\circ_{\mathcal F}$ and ${\mathcal L}_{\mathcal F}$. The authors develop parallel results for the derangement transformation ${\mathcal D}$, obtaining gamma-positive symmetric decompositions for reciprocal derangement polynomials and interlacing structures. By connecting h-polynomials, local h-polynomials, and theta polynomials within the theory of uniform triangulations, they propose unified real-rootedness conjectures that encompass multiple known results and conjectures in the literature, supported by both proofs in key cases (notably barycentric subdivision) and substantial computational evidence. The work significantly bridges permutation statistics, polyhedral subdivisions, and triangulation enumeration, offering a cohesive, generalized perspective on real-rootedness, unimodality, and gamma-positivity in combinatorial settings.

Abstract

The Eulerian transformation is the linear operator on polynomials in one variable with real coefficients which maps the powers of this variable to the corresponding Eulerian polynomials. The derangement transformation is defined similarly. Brändén and Jochemko have conjectured that the Eulerian transforms of a class of polynomials with nonnegative coefficients, which includes those having all their roots in the interval $[-1,0]$, have only real zeros. This conjecture is proven in this paper. More general transformations are introduced in the combinatorial-geometric context of uniform triangulations of simplicial complexes, where Eulerian and derangement transformations arise in the special case of barycentric subdivision, and are shown to have strong unimodality and gamma-positivity properties. General real-rootedness conjectures for these transformations, which unify various results and conjectures in the literature, are also proposed.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Example 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['thm:main']}