发表机构
Kwansei Gakuin University; Toho University（关西学院大学; 东海大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了伪对称单纯反射多胞体的极对偶的Ehrhart多项式具有魔幻正性，并进一步证明伪对称光滑Fano多胞体的魔幻系数严格为正，从而得出其极对偶Ehrhart正且$h^*$-多项式实根且$\u03b3$-正，同时解决了Konoike关于循环对称边多胞体的问题。
AI 中文摘要
受Gal猜想的启发，Ferroni和Higashitani猜想：任何 admitting 一个二次三角剖分的Gorenstein格多胞体的$h^*$-多项式是$\u03b3$-正的。结合预测光滑格多胞体的环面理想具有二次性质的猜想，这表明光滑Gorenstein格多胞体的$h^*$-多项式应当是$\u03b3$-正的。在本文中，我们证明了每个伪对称单纯反射多胞体的极对偶的Ehrhart多项式是魔幻正的。对于伪对称光滑Fano多胞体，我们证明了更强的结论：所有魔幻系数都严格为正。因此，对于每个伪对称单纯反射多胞体，其极对偶是Ehrhart正的，并且具有实根和$\u03b3$-正的$h^*$-多项式。我们还证明了每个循环的对称边多胞体的极对偶的Ehrhart多项式是魔幻正的。这肯定地回答了Konoike的一个问题，他此前曾证明了这一族的部分正性结果。
英文摘要
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.
Comments24 pages
