Cayley decompositions of lattice polytopes and upper bounds for   h^*-polynomials
  We give an effective upper bound on the h^*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. In an appendix, we interpret this result in terms of adjunction theory for toric varieties. 
Christian Haase, Benjamin Nill, Sam Payne