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D.13.1.5 groebnerFan

Procedure from library oldpolymake.lib (see oldpolymake_lib).

Usage:
groebnerFan(f[,#]); f poly, # string

Return:
list, the ith entry of L[1] contains information about the ith cone in the Groebner fan dual to the ith vertex in the Newton polytope of the f
L[1][i][1] = integer matrix representing the inequalities which describe the cone
L[1][i][2] = a list which contains the inequalities represented by L[1][i][1] as a list of strings
L[1][i][3] = 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
L[3] = the Newton polytope of f in the format of the procedure newtonPolytope
L[4] = integer matrix where each row represents the exponet vector of one monomial occuring in the input polynomial

Note:
- if you have already computed the Newton polytope of f then you might want to use the procedure normalFan instead in order to avoid doing costly computation twice
- 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.math.tu-berlin.de/polymake/
- the procedure creates the file /tmp/newtonPolytope.polymake which contains the Newton polytope of f in polymake format and which can be used for further computations with polymake
- it is possible to give as an optional second argument as string which then will be used instead of 'newtonPolytope' in the name of the polymake output file

Example:
 
LIB "oldpolymake.lib";
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 gf=groebnerFan(f);
// the exponent vectors of f are ordered as follows
gf[4];
// the first cone of the groebner fan has the inequalities
gf[1][1][1];
// as a string they look like
gf[1][1][2];
// and it has the extreme rays
print(gf[1][1][3]);
// the linearity space is spanned by
print(gf[2]);
// the vertices of the Newton polytope are:
gf[3][1];
// its dimension is
gf[3][2];
// np[3] contains information how the vertices are connected to each other,
// e.g. the 1st vertex is connected to the 2nd, 3rd and 4th vertex
gf[3][3][1];


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