On the magic positivity of Ehrhart polynomials of dilated polytopes

A polynomial $f(x)$ of degree d is said to be magic positive if all the coefficients of its expansion with respect to the basis ${x^i(x+1)^{d-i}}_{i=0}^d$ are nonnegative. It is known that if $f(x)$ is magic positive, then the polynomial appearing in the numerator of its generating function is real-rooted.

In this talk, we explain that if a polytope is Ehrhart positive, then sufficiently large dilations make its Ehrhart polynomial magic positive, and once it becomes magic positive, it remains so under further dilations. Finally, we investigate how much certain polytopes need to be dilated to make their Ehrhart polynomials magic positive.