Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids

@misc{xie2026realrootedness,
  title = {{Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids}},
  author = {Matthew H. Y. Xie and Philip B. Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07636},
  eprint = {2609.07636},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07636},
}

TY  - UNPB
TI  - Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
AU  - Matthew H. Y. Xie
AU  - Philip B. Zhang
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07636
UR  - https://arxiv.org/abs/2609.07636
AB  - 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.
ER  -