Real-rootedness of augmented Chow polynomials for Cohen–Macaulay posets

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

Background

Augmented Chow polynomials are related to Chow polynomials by the identity G_P(t)=H_{\operatorname{aug}(P)}(t). Because augmentation preserves Cohen–Macaulayness, the general real-rootedness conjecture for Chow polynomials also yields the corresponding unresolved conjecture for augmented Chow polynomials. The paper settles the augmented case for uniform geometric lattices but does not resolve it for all Cohen–Macaulay posets.

References

Since augmenting a poset preserves Cohen-Macaulayness, augmented Chow polynomials are also conjectured to be real rooted for Cohen-Macaulay posets.

— 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”