  We discuss the distribution of the spectrum at infinity of a convenient and
nondegenerate Laurent polynomial (singularity side) and the distribution of the
Newton spectrum of a polytope (Ehrhart theory side). To this end, we study a
hard Lefschetz property for Laurent polynomials and for polytopes and we give
combinatorial criteria for this property to be true. This provides informations
about a conjecture by Katzarkov-Kontsevitch-Pantev.

Algebraic Geometry