# Singular

#### D.13.4.40 tropicalVariety

Procedure from library tropical.lib (see tropical_lib).

Usage:
tropicalVariety(f[,p]); f poly, p optional number
tropicalVariety(I[,p]); I ideal, p optional number

Assume:
I homogeneous, p prime number

Return:
fan, the tropical variety of f resp. I with respect to the trivial valuation or 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; tropicalVariety(I); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 4 ==> ==> DIM ==> 2 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -3 1 1 1 # 0 ==> 1 -3 1 1 # 1 ==> 1 1 -3 1 # 2 ==> 1 1 1 -3 # 3 ==> ==> N_RAYS ==> 4 ==> ==> LINEALITY_SPACE ==> -1 -1 -1 -1 # 0 ==> ==> ORTH_LINEALITY_SPACE ==> 1 -1 0 0 # 0 ==> 1 0 -1 0 # 1 ==> 1 0 0 -1 # 2 ==> ==> F_VECTOR ==> 1 4 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> ==> MAXIMAL_CONES ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> tropicalVariety(I,number(2)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 3 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -2 -1 1 -1 1 # 0 ==> -1 1 -1 1 -1 # 1 ==> 0 -3 1 1 1 # 2 ==> 0 1 -3 1 1 # 3 ==> 0 1 1 -3 1 # 4 ==> 0 1 1 1 -3 # 5 ==> ==> N_RAYS ==> 6 ==> ==> 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 6 5 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 4} ==> {1 3} ==> {1 5} ==> ==> MAXIMAL_CONES ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 4} ==> {1 3} ==> {1 5} ==> tropicalVariety(I,number(3)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 3 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -2 -1 -1 1 1 # 0 ==> -2 1 1 -1 -1 # 1 ==> 0 -3 1 1 1 # 2 ==> 0 1 -3 1 1 # 3 ==> 0 1 1 -3 1 # 4 ==> 0 1 1 1 -3 # 5 ==> ==> N_RAYS ==> 6 ==> ==> 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 6 5 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {1 4} ==> {1 5} ==> ==> MAXIMAL_CONES ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {1 4} ==> {1 5} ==> tropicalVariety(I,number(5)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 3 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -4 -1 -1 -1 3 # 0 ==> 0 -3 1 1 1 # 1 ==> 0 1 -3 1 1 # 2 ==> 0 1 1 -3 1 # 3 ==> 0 1 1 1 -3 # 4 ==> ==> N_RAYS ==> 5 ==> ==> 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 5 4 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==> ==> MAXIMAL_CONES ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==> tropicalVariety(I,number(7)); ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 3 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -1 0 0 0 0 # 0 ==> 0 -3 1 1 1 # 1 ==> 0 1 -3 1 1 # 2 ==> 0 1 1 -3 1 # 3 ==> 0 1 1 1 -3 # 4 ==> ==> N_RAYS ==> 5 ==> ==> 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 5 4 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==> ==> MAXIMAL_CONES ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {0 4} ==>