That polyhedron is unbounded; so the function says it cannot count the lattice points. Notice the extra inequality "1 0 0" (which amounts to 1 >= 0) that was added by polymake to your input. This is useful for combinatorial reasons: https://polymake.org/doku.php/user_guide/tutorials/coordi...
The following variant of the tutorial code is more generic and thus probably even better suited for your needs: polymake 'my $c=load("c3.poly"); print "$_\n", $c->$_, "\n\n" for "N_VERTICES", "N_FACETS";' You can add as many properties as you like.
Yes, inequalities and equalities are kept seperate. You can use them in the way you described as input (by specifying INEQUALITIES and maybe EQUATIONS), but FACETS and AFFINE_HULL give irredundant output.
I am glad you found something useful in polymake. Let me point out that polymake scripts differ slightly from general perl scripts. They should start out with something like use application "polytope"; to set the scope (i.e., the active application). polymake's determinant function lives i...
Is there a way to tell polymake to use the polymake det function? This is impossible to answer without seeing your script. I do not know about pdl. That is, if you would post that script, it might require a bit of explanation. If not, is there is an efficient way to compute the multiplicity of a se...
To get an idea I would recommend to look at code in polymake's source, e.g., normal_fan.cc. While that is about a different big object type, its use is similar.