  We introduce a canonical strip hypothesis for Fano varieties. We show that
the canonical strip hypothesis for a Fano variety implies that the zeros of the
Hilbert polynomial of embedded Calabi--Yau and general type hypersurfaces are
located on a vertical line. This extends, in particular, Villegas's `polynomial
RH' for intersections in projective spaces to the case of CY and general type
hyperplane sections in Grassmannians. We state a few conjectures on the Ehrhart
polynomials of certain fan polytopes.
