Let $M$ be a (loopless) matroid of rank $r$.


The *characteristic polynomial* $\chi\_M(x)$ is defined by $\chi\_M(x)=\sum\_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(M)$ is the lattice of flats of $M$ and $\mu$ its Möbius function. It is known that the signs of the characteristic polynomial alternate, and the so-called *Whitney numbers of the 1st kind* $\omega\_i$ are defined by $\chi\_M(x) = \sum\_{i=0}^{r} (-1)^i \omega\_i x^i$.


In [Hodge theory for combinatorial geometries](https://doi.org/10.4007/annals.2018.188.2.1%0A), affirming a long-standing conjecture of Rota and others, Adiprasito-Huh-Katz (AHK) showed that these $\omega\_i$ form a *log-concave sequence*, i.e., $\omega\_i^2 \geq \omega\_{i-1}\omega\_{i+1}$.


Actually, they proved something stronger. It is known that $(1-x)$ is always a factor of $\chi\_M(x)$, and the *reduced Whitney numbers* $\overline{\omega}\_i$ are thus defined by $\chi\_M(x)/(1-x)=\sum\_{i=0}^{r-1} (-1)^i \overline{\omega}\_i x^i$. AHK showed that in fact the reduced Whitney numbers form a log-concave sequence: $\overline{\omega}\_i^2 \geq \overline{\omega}\_{i-1}\overline{\omega}\_{i+1}$ (which implies log-concavity of the $\omega\_i$ by a straightforward argument).


In fact, an easy reduction using the *truncation* of the matroid $M$ shows that it is enough to prove log-concavity of the $\overline{\omega}\_i$ in the "last-spot": i.e., that $\overline{\omega}\_{r-2}^2 \geq \overline{\omega}\_{r-3}\cdot\overline{\omega}\_{r-1}$.


To prove this, AHK use the *Chow ring* $A(M) = A^0(M)\oplus A^1(M)\oplus \cdots\oplus A^{r-1}(M)$ of the matroid $M$: they show that $\overline{\omega}\_k=\langle \alpha^{r-1-k}\beta^{k}\rangle$ for certain linear elements $\alpha,\beta\in A^1(M)$ of the Chow ring, where $\langle \cdot \rangle\colon A^{r-1}(M)\to \mathbb{R}$ is the canonical *degree map* isomorphism; and they deduce $\overline{\omega}\_{r-2}^2 \geq \overline{\omega}\_{r-3}\cdot\overline{\omega}\_{r-1}$ from the so-called *Kähler package* for $A(M)$, in particular, the *Hodge-Riemann relations*, which imply that the relevant $2\times 2$ determinant is nonpositive.


**Question**: From the work of AHK (or elsewhere) is it possible to deduce when (i.e. for which matroids) we have an equality $\overline{\omega}\_{r-2}^2 = \overline{\omega}\_{r-3}\cdot\overline{\omega}\_{r-1}$? (In other words, when the determinant of the $2\times 2$ HR relations matrix is $0$ rather than negative?)



Log-concavity of matroids: characterization of equality?
I think the following shows it's never possible for there to be equality.


Indeed, Ardila-Denham-Huh <https://arxiv.org/abs/2004.13116> recently showed for any matroid $M$ that $T\_M(x,0)$ has log-concave coefficient sequence, and hence obviously $\frac{1}{x}T\_M(x,0)$ has log-concave coefficient sequence (Note that $T\_M$ always has $0$ constant term except when $M=\emptyset$).


Now, Lenz in Lemma 4.2 of <https://arxiv.org/abs/1106.2944> showed that if $f(x)$ has non-negative log-concave coefficient sequence then $f(x+1)$ has strictly log-concave coefficient sequence.


Taking $f(x)=\frac{1}{x}T\_M(x,0)$, we have $f(x+1)=\frac{1}{1+x}T\_M(1+x,0)$ has strictly log-concave coefficient sequence. But this is exactly the reduced characteristic polynomial with all coefficients made to have positive signs.


