Fano geography and the polynomial transform

Given a Fano or a general type variety, one can study the position of the roots of its Hilbert polynomial H(z) with respect to the anticanonical class. By intersection theory, its coefficients are simple expressions in the Chern numbers. The well-known Routh-Hurwitz stability criterion says that for H(z) to have all roots in a given vertical strip, or on a line, a system of polynomial inequalities in Chern numbers must be satisfied. Work by Manivel and others establishes vertical strip properties for certain classes of Fano varieties.

Now, the generating function of the sequence P(n) of values of any polynomial is rational. Its numerator is another polynomial, called the transform of P. Motivated by the above considerations I will discuss some polynomials arising from dual lattice polytopes, their transforms, and their role in mirror symmetry.