Singular

D.13.2.1 polymakePolytope

Usage:

polymakePolytope(points); polytope intmat

Assume:

each row of points gives the coordinates of a lattice point of a polytope with their affine coordinates as given by the output of secondaryPolytope

Purpose:

the procedure calls polymake to compute the vertices of the polytope as well as its dimension and information on its facets

Return:

list, L with four entries
L[1] : an integer matrix whose rows are the coordinates of vertices of the polytope
L[2] : the dimension of the polytope
L[3] : a list whose ith entry explains to which vertices the ith vertex of the Newton polytope is connected
-- i.e. L[3][i] is an integer vector and an entry k in there means that the vertex L[1][i] is connected to the vertex L[1][k]
L[4] : an matrix of type bigintmat whose rows multiplied by (1,var(1),...,var(nvar)) give a linear system of equations describing the affine hull of the polytope,
i.e. the smallest affine space containing the polytope

Note:

- for its computations the procedure calls the program polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt; it therefore is necessary that this program is installed in order to use this procedure;
see http://www.polymake.org/
- note that in the vertex edge graph we have changed the polymake convention which starts indexing its vertices by zero while we start with one !

LIB "polymake.lib";
==> Welcome to polymake version
==> Copyright (c) 1997-2015
==> Ewgenij Gawrilow, Michael Joswig (TU Darmstadt)
==> http://www.polymake.org
// the lattice points of the unit square in the plane
list points=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
// the secondary polytope of this lattice point configuration is computed
intmat secpoly=secondaryPolytope(points)[1];
==> Evaluating Commandline Options ...
==> ... done.
==> 0
==> 0
list np=polymakePolytope(secpoly);
==> polymake: used package ppl
==>   The Parma Polyhedra Library (PPL): A C++ library for convex polyhedra
==>   and other numerical abstractions.
==>   http://www.cs.unipr.it/ppl/
==> 
// the vertices of the secondary polytope are:
np[1];
==> 2,1,1,2,
==> 1,2,2,1
// its dimension is
np[2];
==> 1
// np[3] contains information how the vertices are connected to each other,
// e.g. the first vertex (number 0) is connected to the second one
np[3][1];
==> 2
// the affine hull has the equation
ring r=0,x(1..4),dp;
matrix M[5][1]=1,x(1),x(2),x(3),x(4);
intmat(np[4])*M;
==> _[1,1]=-x(3)-x(4)+3
==> _[2,1]=x(1)-x(4)
==> _[3,1]=x(2)-x(3)