Singular Manual: polymakePolytope

| #### D.13.2.1 polymakePolytope Procedure from library `polymake.lib` (see polymake_lib). 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 ! Example:
    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)

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