The chain polynomial of a finite lattice \( \mathcal{L} \) is given by \( p_\mathcal{L} (x) = \sum_{k \geq 0} c_k(\mathcal{L})x^k \), where \( c_k(\mathcal{L}) \) is the number of chains of length \( k \) in \( \mathcal{L} \). The chain polynomials of posets in several important classes have been proven to be realrooted, for example, face lattices of simplicial and cubical polytopes, and (3+1)free posets, proved by Brenti and Welker, Athanasiadis, and Stanley, respectively. However, Stembridge presented some distributive lattices that do not have realrooted polynomials. Recently, Athanasiadis and KalampogiaEvangelinou conjectured that the chain polynomials of geometric lattices are always realrooted. We present a sufficient theorem that proves this conjecture for the lattice of flats of paving matroids and generalized paving matroids. We also apply this theorem to other posets to verify that their chain polynomials are also realrooted. This is a joint work with Petter Brändén.
