A relation between number of integral points, volumes of faces and   degree of the discriminant of smooth lattice polytopes
  We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than $\lceil \frac {\dim P} 2 \rceil$. 
Alicia Dickenstein, Benjamin Nill and Mich\`ele Vergne
math.CO math.AG