Singular

D.13.2.4 normalFanL

Usage:

normalFanL (vert,aff,graph,rays,[,#]); vert,aff intmat, graph list, rays int, # string

Assume:

- vert is an integer matrix whose rows are the coordinate of the vertices of a convex lattice polytope;
- aff describes the affine hull of this polytope, i.e. the smallest affine space containing it, in the following sense: denote by n the number of columns of vert, then multiply aff by (1,x(1),...,x(n)) and set the resulting terms to zero in order to get the equations for the affine hull;
- the ith entry of graph is an integer vector describing to which vertices the ith vertex is connected, i.e. a k as entry means that the vertex vert[i] is connected to vert[k];
- the integer rays is either one (if the extreme rays should be computed) or zero (otherwise)

Return:

list, the ith entry of L[1] contains information about the cone in the normal fan dual to the ith vertex of the polytope
L[1][i][1] = integer matrix representing the inequalities which describe the cone dual to the ith vertex
L[1][i][2] = a list which contains the inequalities represented by L[i][1] as a list of strings, where we use the
variables x(1),...,x(n)
L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix whose rows are the extreme rays
of the cone
L[2] = is an integer matrix whose rows span the linearity space of the fan, i.e. the linear space which is contained in each cone

Note:

- the procedure calls for its computation polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, so it only works if polymake is installed;
see http://www.polymake.org/
- in the optional argument # it is possible to hand over other names for the variables to be used -- be careful, the format must be correct and that is not tested, e.g. if you want the variable names to be u00,u10,u01,u11 then you must hand over the string u11,u10,u01,u11

LIB "polymake.lib";
==> Welcome to polymake version
==> Copyright (c) 1997-2015
==> Ewgenij Gawrilow, Michael Joswig (TU Darmstadt)
==> http://www.polymake.org
ring r=0,(x,y,z),dp;
matrix M[4][1]=1,x,y,z;
poly f=y3+x2+xy+2xz+yz+z2+1;
// the Newton polytope of f is
list np=newtonPolytopeP(f);
==> 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 Groebner fan of f, i.e. the normal fan of the Newton polytope
list gf=normalFanL(np[1],np[4],np[3],1,"x,y,z");
// the number of cones in the Groebner fan of f is:
size(gf[1]);
==> 4
// the inequalities of the first cone as matrix are:
print(gf[1][1][1]);
==> -2, 0, 0,
==>  0,-3, 0,
==>  0, 0,-2
// the inequalities of the first cone as string are:
print(gf[1][1][2]);
==> [1]:
==>    0 > 2x
==> [2]:
==>    0 > 3y
==> [3]:
==>    0 > 2z
// the rows of the following matrix are the extreme rays of the first cone:
print(gf[1][1][3]);
==> -1, 0, 0,
==>  0,-1, 0,
==>  0, 0,-1
// each cone contains the linearity space spanned by:
print(gf[2]);
==> 0,0,0