  We present a polynomial time algorithm to compute any fixed number of the
highest coefficients of the Ehrhart quasi-polynomial of a rational simplex.
Previously such algorithms were known for integer simplices and for rational
polytopes of a fixed dimension. The algorithm is based on the formula relating
the kth coefficient of the Ehrhart quasi-polynomial of a rational polytope to
volumes of sections of the polytope by affine lattice subspaces parallel to
k-dimensional faces of the polytope. We discuss possible extensions and open
questions.
