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A weighted inequality for matroid flats and log-concavity of Whitney numbers

  • Yuxian Dong
  • Yanxin Liu
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Abstract

Using the Hodge-Riemann relations for matroid Chow rings, we establish a weighted inequality for flats of a loopless matroid. As applications, we prove the points-lines-planes conjecture and Brylawski's conjecture on the ultra-log-concavity of Whitney numbers of the first kind. We also characterize the cases of equality in both inequalities.

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Cite

BibTeXRIS
BibTeX
@misc{dong2026weighted,
  title = {{A weighted inequality for matroid flats and log-concavity of Whitney numbers}},
  author = {Yuxian Dong and Yanxin Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29893},
  eprint = {2610.29893},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29893},
}
RIS
TY  - UNPB
TI  - A weighted inequality for matroid flats and log-concavity of Whitney numbers
AU  - Yuxian Dong
AU  - Yanxin Liu
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29893
UR  - https://arxiv.org/abs/2610.29893
AB  - Using the Hodge-Riemann relations for matroid Chow rings, we establish a weighted inequality for flats of a loopless matroid. As applications, we prove the points-lines-planes conjecture and Brylawski's conjecture on the ultra-log-concavity of Whitney numbers of the first kind. We also characterize the cases of equality in both inequalities.
ER  -