Singular

D.13.4.42 groebnerComplex

Procedure from library `tropical.lib` (see tropical_lib).

Usage:
groebnerComplex(f,p); f poly, p number
groebnerComplex(I,p); I ideal, p number

Assume:
I homogeneous, p prime number

Return:
fan, the Groebner complex of f resp. I with respect to the p-adic valuation

Note:
set printlevel=1 for output during traversal

Example:
 ```LIB "tropical.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,w),dp; ideal I = x-2y+3z,3y-4z+5w; groebnerComplex(I,number(2)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 5 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -2 -1 1 -1 1 # 0 ==> -1 1 -1 1 -1 # 1 ==> 0 -3 1 1 1 # 2 ==> 0 -1 -1 -1 3 # 3 ==> 0 -1 -1 3 -1 # 4 ==> 0 -1 3 -1 -1 # 5 ==> 0 1 -3 1 1 # 6 ==> 0 1 1 -3 1 # 7 ==> 0 1 1 1 -3 # 8 ==> 0 3 -1 -1 -1 # 9 ==> ==> N_RAYS ==> 10 ==> ==> LINEALITY_SPACE ==> 0 -1 -1 -1 -1 # 0 ==> ==> ORTH_LINEALITY_SPACE ==> -1 0 0 0 0 # 0 ==> 0 1 -1 0 0 # 1 ==> 0 1 0 -1 0 # 2 ==> 0 1 0 0 -1 # 3 ==> ==> F_VECTOR ==> 1 10 23 22 8 ==> ==> SIMPLICIAL ==> 0 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {6} ==> {7} ==> {8} ==> {9} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 5} ==> {0 7} ==> {1 4} ==> {1 6} ==> {1 8} ==> {1 9} ==> {2 3} ==> {2 4} ==> {2 5} ==> {2 7} ==> {3 6} ==> {4 6} ==> {3 7} ==> {4 8} ==> {5 7} ==> {5 8} ==> {6 8} ==> {6 9} ==> {7 9} ==> {8 9} ==> {0 1 2 4} # Dimension 4 ==> {0 1 3 6} ==> {0 1 5 8} ==> {0 1 7 9} ==> {0 2 3} ==> {0 2 5} ==> {0 2 7} ==> {0 3 7} ==> {0 5 7} ==> {1 4 6} ==> {1 4 8} ==> {1 6 8} ==> {1 6 9} ==> {1 8 9} ==> {2 3 4 6} ==> {2 4 5 8} ==> {2 3 7} ==> {2 5 7} ==> {4 6 8} ==> {3 6 7 9} ==> {5 7 8 9} ==> {6 8 9} ==> {0 1 2 3 4 6} # Dimension 5 ==> {0 1 2 4 5 8} ==> {0 1 3 6 7 9} ==> {0 1 5 7 8 9} ==> {0 2 3 7} ==> {0 2 5 7} ==> {1 4 6 8} ==> {1 6 8 9} ==> ==> MAXIMAL_CONES ==> {0 1 2 3 4 6} # Dimension 5 ==> {0 1 2 4 5 8} ==> {0 1 3 6 7 9} ==> {0 1 5 7 8 9} ==> {0 2 3 7} ==> {0 2 5 7} ==> {1 4 6 8} ==> {1 6 8 9} ==> groebnerComplex(I,number(3)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 5 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -2 -1 -1 1 1 # 0 ==> -2 1 1 -1 -1 # 1 ==> 0 -3 1 1 1 # 2 ==> 0 -1 -1 -1 3 # 3 ==> 0 -1 -1 3 -1 # 4 ==> 0 -1 3 -1 -1 # 5 ==> 0 1 -3 1 1 # 6 ==> 0 1 1 -3 1 # 7 ==> 0 1 1 1 -3 # 8 ==> 0 3 -1 -1 -1 # 9 ==> ==> N_RAYS ==> 10 ==> ==> LINEALITY_SPACE ==> 0 -1 -1 -1 -1 # 0 ==> ==> ORTH_LINEALITY_SPACE ==> -1 0 0 0 0 # 0 ==> 0 1 -1 0 0 # 1 ==> 0 1 0 -1 0 # 2 ==> 0 1 0 0 -1 # 3 ==> ==> F_VECTOR ==> 1 10 23 22 8 ==> ==> SIMPLICIAL ==> 0 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {6} ==> {7} ==> {8} ==> {9} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==> {0 6} ==> {1 5} ==> {1 7} ==> {1 8} ==> {1 9} ==> {2 3} ==> {2 4} ==> {2 5} ==> {2 6} ==> {3 6} ==> {4 6} ==> {3 7} ==> {4 8} ==> {5 7} ==> {5 8} ==> {7 8} ==> {6 9} ==> {7 9} ==> {8 9} ==> {0 1 2 5} # Dimension 4 ==> {0 1 3 7} ==> {0 1 4 8} ==> {0 1 6 9} ==> {0 2 3} ==> {0 2 4} ==> {0 2 6} ==> {0 3 6} ==> {0 4 6} ==> {1 5 7} ==> {1 5 8} ==> {1 7 8} ==> {1 7 9} ==> {1 8 9} ==> {2 3 5 7} ==> {2 4 5 8} ==> {2 3 6} ==> {2 4 6} ==> {5 7 8} ==> {3 6 7 9} ==> {4 6 8 9} ==> {7 8 9} ==> {0 1 2 3 5 7} # Dimension 5 ==> {0 1 2 4 5 8} ==> {0 1 3 6 7 9} ==> {0 1 4 6 8 9} ==> {0 2 3 6} ==> {0 2 4 6} ==> {1 5 7 8} ==> {1 7 8 9} ==> ==> MAXIMAL_CONES ==> {0 1 2 3 5 7} # Dimension 5 ==> {0 1 2 4 5 8} ==> {0 1 3 6 7 9} ==> {0 1 4 6 8 9} ==> {0 2 3 6} ==> {0 2 4 6} ==> {1 5 7 8} ==> {1 7 8 9} ==> groebnerComplex(I,number(5)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 5 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -4 -1 -1 -1 3 # 0 ==> 0 -3 1 1 1 # 1 ==> 0 -1 -1 -1 3 # 2 ==> 0 -1 -1 3 -1 # 3 ==> 0 -1 3 -1 -1 # 4 ==> 0 1 -3 1 1 # 5 ==> 0 1 1 -3 1 # 6 ==> 0 1 1 1 -3 # 7 ==> 0 3 -1 -1 -1 # 8 ==> ==> N_RAYS ==> 9 ==> ==> LINEALITY_SPACE ==> 0 -1 -1 -1 -1 # 0 ==> ==> ORTH_LINEALITY_SPACE ==> -1 0 0 0 0 # 0 ==> 0 1 -1 0 0 # 1 ==> 0 1 0 -1 0 # 2 ==> 0 1 0 0 -1 # 3 ==> ==> F_VECTOR ==> 1 9 20 18 6 ==> ==> SIMPLICIAL ==> 0 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {6} ==> {7} ==> {8} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==> {0 5} ==> {0 6} ==> {0 7} ==> {0 8} ==> {1 2} ==> {1 3} ==> {1 4} ==> {2 5} ==> {3 5} ==> {2 6} ==> {3 7} ==> {4 6} ==> {4 7} ==> {5 8} ==> {6 8} ==> {7 8} ==> {0 1 2} # Dimension 4 ==> {0 1 3} ==> {0 1 4} ==> {0 2 5} ==> {0 3 5} ==> {0 2 6} ==> {0 3 7} ==> {0 4 6} ==> {0 4 7} ==> {0 5 8} ==> {0 6 8} ==> {0 7 8} ==> {1 2 3 5} ==> {1 2 4 6} ==> {1 3 4 7} ==> {2 5 6 8} ==> {3 5 7 8} ==> {4 6 7 8} ==> {0 1 2 3 5} # Dimension 5 ==> {0 1 2 4 6} ==> {0 1 3 4 7} ==> {0 2 5 6 8} ==> {0 3 5 7 8} ==> {0 4 6 7 8} ==> ==> MAXIMAL_CONES ==> {0 1 2 3 5} # Dimension 5 ==> {0 1 2 4 6} ==> {0 1 3 4 7} ==> {0 2 5 6 8} ==> {0 3 5 7 8} ==> {0 4 6 7 8} ==> groebnerComplex(I,number(7)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 5 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -1 0 0 0 0 # 0 ==> 0 -3 1 1 1 # 1 ==> 0 -1 -1 -1 3 # 2 ==> 0 -1 -1 3 -1 # 3 ==> 0 -1 3 -1 -1 # 4 ==> 0 1 -3 1 1 # 5 ==> 0 1 1 -3 1 # 6 ==> 0 1 1 1 -3 # 7 ==> 0 3 -1 -1 -1 # 8 ==> ==> N_RAYS ==> 9 ==> ==> LINEALITY_SPACE ==> 0 -1 -1 -1 -1 # 0 ==> ==> ORTH_LINEALITY_SPACE ==> -1 0 0 0 0 # 0 ==> 0 1 -1 0 0 # 1 ==> 0 1 0 -1 0 # 2 ==> 0 1 0 0 -1 # 3 ==> ==> F_VECTOR ==> 1 9 20 18 6 ==> ==> SIMPLICIAL ==> 0 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {6} ==> {7} ==> {8} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==> {0 5} ==> {0 6} ==> {0 7} ==> {0 8} ==> {1 2} ==> {1 3} ==> {1 4} ==> {2 5} ==> {3 5} ==> {2 6} ==> {3 7} ==> {4 6} ==> {4 7} ==> {5 8} ==> {6 8} ==> {7 8} ==> {0 1 2} # Dimension 4 ==> {0 1 3} ==> {0 1 4} ==> {0 2 5} ==> {0 3 5} ==> {0 2 6} ==> {0 3 7} ==> {0 4 6} ==> {0 4 7} ==> {0 5 8} ==> {0 6 8} ==> {0 7 8} ==> {1 2 3 5} ==> {1 2 4 6} ==> {1 3 4 7} ==> {2 5 6 8} ==> {3 5 7 8} ==> {4 6 7 8} ==> {0 1 2 3 5} # Dimension 5 ==> {0 1 2 4 6} ==> {0 1 3 4 7} ==> {0 2 5 6 8} ==> {0 3 5 7 8} ==> {0 4 6 7 8} ==> ==> MAXIMAL_CONES ==> {0 1 2 3 5} # Dimension 5 ==> {0 1 2 4 6} ==> {0 1 3 4 7} ==> {0 2 5 6 8} ==> {0 3 5 7 8} ==> {0 4 6 7 8} ==> ```