Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes

Abstract

Motivated by Gal's conjecture, Ferroni and Higashitani conjectured that the h^*-polynomial of any Gorenstein lattice polytope admitting a quadratic triangulation is γ-positive. Together with conjectures predicting quadratic properties of toric ideals of smooth lattice polytopes, this suggests that smooth Gorenstein lattice polytopes should have γ-positive h^*-polynomials. In this paper, we prove that the Ehrhart polynomial of the polar dual of every pseudo-symmetric simplicial reflexive polytope is magic positive. For pseudo-symmetric smooth Fano polytopes, we prove the stronger statement that all magic coefficients are strictly positive. Consequently, for every pseudo-symmetric simplicial reflexive polytope, its polar dual is Ehrhart positive and has a real-rooted and γ-positive h^*-polynomial. We also prove that the Ehrhart polynomial of the polar dual of the symmetric edge polytope of every cycle is magic positive. This gives an affirmative answer to a question of Konoike, who had previously proved partial positivity results for this family.

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BibTeX RIS

BibTeX

@misc{ohsugi2026magic,
  title = {{Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes}},
  author = {Hidefumi Ohsugi and Akiyoshi Tsuchiya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06419},
  eprint = {2610.06419},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06419},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes
AU  - Hidefumi Ohsugi
AU  - Akiyoshi Tsuchiya
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06419
UR  - https://arxiv.org/abs/2610.06419
AB  - Motivated by Gal's conjecture, Ferroni and Higashitani conjectured that the h^*-polynomial of any Gorenstein lattice polytope admitting a quadratic triangulation is γ-positive. Together with conjectures predicting quadratic properties of toric ideals of smooth lattice polytopes, this suggests that smooth Gorenstein lattice polytopes should have γ-positive h^*-polynomials. In this paper, we prove that the Ehrhart polynomial of the polar dual of every pseudo-symmetric simplicial reflexive polytope is magic positive. For pseudo-symmetric smooth Fano polytopes, we prove the stronger statement that all magic coefficients are strictly positive. Consequently, for every pseudo-symmetric simplicial reflexive polytope, its polar dual is Ehrhart positive and has a real-rooted and γ-positive h^*-polynomial. We also prove that the Ehrhart polynomial of the polar dual of the symmetric edge polytope of every cycle is magic positive. This gives an affirmative answer to a question of Konoike, who had previously proved partial positivity results for this family.
ER  -

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