Chow polynomials of rank-uniform labeled posets

We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni–Schröter, Huh–Stevens, and Athanasiadis–Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert–Poincaré series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert–Poincaré series of the Stanley–Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory.