The Volume of a Lattice

In the area of automatic parallelization of programs, restructura-tion of nested loops or parallelization of systems of aane recurrence equations use classically a geometrical model based on convex polyhedra of lattices. Since these algorithms are speciied for parametric sized calculation domains, para-metric lattice polyhedra have to be considered. Computing on such domains is a grand challenge in today's discrete mathematics research. In particular, the number of lattice points in a polyhedron has several useful meanings in automatic parallelization. At the end of the 50's, E. Ehrhart introduced the concept of homothetic polyhedra suprisingly tting so well to our context. We recall in this paper Ehrhart's powerful results allowing to compute the number of lattice points in integral or rational polyhedra of any dimension which may be non-convex, and apply these results to the computation of the number of processors resulting from linear allocations of the calculation domain, and of the potential parallelism occurring in parallel algorithms.