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

Source 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.