Real-rootedness of matroid Chow polynomials

Prove that the Chow polynomial H_M(t) of every matroid M has only real roots.

Background

For a matroid M, the Chow polynomial H_M(t) is the Hilbert–Poincaré series of its Chow ring. The paper notes that matroidal Hodge theory implies that H_M(t) is monic and palindromic with unimodal coefficients. It further records a conjecture asserting the stronger property that all roots are real; this is presented as a conjecture for general matroids, although the paper notes that it has been confirmed for all uniform matroids.

References

It is for instance conjectured that $H_M(t)$ has only real roots, and this was recently confirmed for all uniform matroids.

— Chow groups of braid matroids and relative maps to $\mathbb{P}^1$  (2504.19829 - Kannan et al., 28 Apr 2025) in Section 1, subsection “Matroid-theoretic context”