  The purpose of this paper is to give an explicit formula which allows one to
compute the dimension of the cohomology groups of the sheaf $\Omega_{\P}^p(D)$
of p-th differential forms of Zariski twisted by an ample invertible sheaf on a
complete simplicial toric variety. The formula involves some combinatorial sums
of integer points over all faces of the support polytope for ${\O_X}(D)$. We
also introduce a new combinatorial object, the so-called p-th Hilbert-Erhart
polynomial, which generalizes the usual notion and behaves similar. Namely,
there exists a generalization of the inversion law for a usual Hilbert-Erhart
polynomial. Some applications of the Bott formula are discussed.
