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D.13.2.5 groebnerFanP
Procedure from library polymake.lib (see polymake_lib).
- Usage:
- groebnerFanP(f); f poly
- 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
newtonPolytopeP
L[4] = integer matrix where each row represents the exponent
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 normalFanL 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.polymake.org/
Example:
| | 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 gf=groebnerFanP(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 exponent vectors of f are ordered as follows
gf[4];
==> 0,3,0,
==> 2,0,0,
==> 1,1,0,
==> 1,0,1,
==> 0,1,1,
==> 0,0,2,
==> 0,0,0
// the first cone of the groebner fan has the inequalities
gf[1][1][1];
==> -2, 0, 0,
==> 0,-3, 0,
==> 0, 0,-2
// as a string they look like
gf[1][1][2];
==> [1]:
==> 0 > 2x
==> [2]:
==> 0 > 3y
==> [3]:
==> 0 > 2z
// and it has the extreme rays
print(gf[1][1][3]);
==> -1, 0, 0,
==> 0,-1, 0,
==> 0, 0,-1
// the linearity space is spanned by
print(gf[2]);
==> 0,0,0
// the vertices of the Newton polytope are:
gf[3][1];
==> 0,0,0,
==> 2,0,0,
==> 0,3,0,
==> 0,0,2
// its dimension is
gf[3][2];
==> 3
// 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];
==> 2,3,4
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