  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.
