Real-rootedness of Chow polynomials for Cohen–Macaulay posets

Prove that the Chow polynomial H_P(t) of every Cohen–Macaulay bounded and graded poset P is real-rooted.

Background

The paper studies Chow polynomials H_P(t) associated with bounded and graded posets. Prior work established positivity, unimodality, and γ-positivity for these polynomials under Cohen–Macaulayness, but the stronger property that all zeros are real is not established in general. The paper proves real-rootedness for uniform geometric lattices and maximal ranked posets, leaving the stated Cohen–Macaulay-poset conjecture unresolved.

References

In addition to that, it is \gamma-positive for Cohen-Macaulay posets Theorem~1.4 and is conjectured to be real-rooted for every Cohen-Macaulay poset Conjecture~1.5.

— Chow polynomials of uniform matroids are real-rooted  (2501.07364 - Brändén et al., 13 Jan 2025) in Section 2, subsection “Characteristic Chow polynomials of graded bounded posets”