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PreprintCC BY 4.0arXiv

Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart h*-polynomials

  • Houshan Fu
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Abstract

We prove that the Ehrhart h*-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and (4,a,a+29), for all sufficiently large integers a, have h*-polynomials that are not real-rooted. These counterexamples disprove Ferroni's real-rootedness conjecture. Their duals are cycle matroids of theta graphs and have the same h*-polynomials. Nevertheless, every matroid of rank two or corank two has a positive h*-coefficient sequence that is ultra log-concave of order equal to the polynomial's degree. In particular, the unimodality conjecture holds in both cases.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026realrootedness,
  title = {{Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart $h^*$-polynomials}},
  author = {Houshan Fu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37439},
  eprint = {2609.37439},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37439},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart h^*-polynomials
AU  - Houshan Fu
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37439
UR  - https://arxiv.org/abs/2609.37439
AB  - We prove that the Ehrhart h^*-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and (4,a,a+29), for all sufficiently large integers a, have h^*-polynomials that are not real-rooted. These counterexamples disprove Ferroni's real-rootedness conjecture. Their duals are cycle matroids of theta graphs and have the same h^*-polynomials. Nevertheless, every matroid of rank two or corank two has a positive h^*-coefficient sequence that is ultra log-concave of order equal to the polynomial's degree. In particular, the unimodality conjecture holds in both cases.
ER  -