Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids
Year: 2026
Type: preprint
Abstract: We prove that the $Z$-polynomial of every sparse paving matroid has only negative real zeros, confirming the real-rootedness conjecture of Proudfoot, Xu, and Young for this class. We also prove that, for each fixed positive corank, the $Z$-polynomials of uniform matroids in consecutive ranks strictly interlace. Our approach is to prove real-rootedness of the corresponding $γ$-polynomials for sparse paving matroids and their strict interlacing for uniform matroids, and then transfer both properties to the corresponding $Z$-polynomials.
Source: arXiv (Cornell University)
Authors: Xie, Matthew H. Y., Zhang, Philip B.
Institutions: Tianjin University of Technology, Tianjin Normal University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6545402480, G6545595343, G7610362479
OpenAlex ID: W7212032052