Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart $h^{\ast }$-polynomials

We prove that the Ehrhart $h^{\ast }$-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and $(4,a,a+29)$, for all sufficiently large integers $a$, have $h^{\ast }$-polynomials that are not real-rooted. These counterexamples disprove Ferroni’s real-rootedness conjecture. Their duals are cycle matroids of theta graphs and have the same $h^{\ast }$-polynomials. Nevertheless, every matroid of rank two or corank two has a positive $h^{\ast }$-coefficient sequence that is ultra log-concave of order equal to the polynomial’s degree. In particular, the unimodality conjecture holds in both cases.