What can LattE compute?

In the following we list the operations that LattE v1.1 can perform on bounded convex polyhedra (more commonly referred to as polytopes). For the reader's convenience, we already include the basic commands to actually do the tasks. Let us assume that a description of a polytope $P$ is given in the file ``fileName'' (see Section 3 for format) and that a cost vector is specified in the file ``fileName.cost'' (needed for the optimization part, see Section 3 for format).

Tasks performed by LattE v1.1:

1. Count the number of lattice points in $P$ .

        ./count fileName
   

2. Count the number of lattice points in $nP$ , the dilation of $P$ by the integer factor $n$ .

        ./count dil n fileName
   

3. Calculate a rational function that encodes the Ehrhart series associated with the polytope. By definition, the $n$ -th coefficient in the Ehrhart series equals the number of lattice points in $nP$ . For more details on Ehrhart counting functions see, for example, Chapter 4 of [9].

        ./ehrhart fileName
   

4. Calculate the first $n+1$ terms of the Ehrhart series associated with the polytope.

        ./ehrhart n fileName
   

5. Maximize or minimize a given linear function of the lattice points in $P$ .

        ./maximize fileName
        ./minimize fileName
   

In addition to these basic functions, there are more specific calls to LattE. For example to use the homogenized Barvinok algorithm instead of the original one in order to count the lattice points. These details will be explained in Section 4.