# Singular

#### D.16.1.8 tropicalVariety

`Syntax:`
`tropicalVariety(` poly g `)`
`tropicalVariety(` poly g, number p `)`
`tropicalVariety(` ideal I `)`
`tropicalVariety(` ideal I, number p `)`
`Assume:`
I homogeneous, ground field are the rational numbers, p prime
`Type:`
fan
`Purpose:`
the tropical variety of g or the tropical variety of I, either without or with p-adic valuation
`Note:`
set printlevel > 0 for status updates on the computation
`Example:`
 ```LIB "gfanlib.so"; ring r = 0,(x,y),dp; poly g1 = x+y+1; fan f1 = tropicalVariety(g1); f1; // tropical line with vertex at (0,0) ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 2 ==> ==> DIM ==> 1 ==> ==> LINEALITY_DIM ==> 0 ==> ==> RAYS ==> -1 0 # 0 ==> 0 -1 # 1 ==> 1 1 # 2 ==> ==> N_RAYS ==> 3 ==> ==> LINEALITY_SPACE ==> ==> ORTH_LINEALITY_SPACE ==> -1 0 # 0 ==> 0 -1 # 1 ==> ==> F_VECTOR ==> 1 3 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 0 ==> {0} # Dimension 1 ==> {1} ==> {2} ==> ==> MAXIMAL_CONES ==> {0} # Dimension 1 ==> {1} ==> {2} ==> ==> MULTIPLICITIES ==> 1 # Dimension 1 ==> 1 ==> 1 ==> poly g2 = 2x+y+4; fan f2 = tropicalVariety(g2,number(2)); f2; // tropical line with vertex at (0,1) ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 3 ==> ==> DIM ==> 2 ==> ==> LINEALITY_DIM ==> 0 ==> ==> RAYS ==> -1 -1 -2 # 0 ==> 0 -1 0 # 1 ==> 0 0 -1 # 2 ==> 0 1 1 # 3 ==> ==> N_RAYS ==> 4 ==> ==> LINEALITY_SPACE ==> ==> ORTH_LINEALITY_SPACE ==> -1 0 0 # 0 ==> 0 -1 0 # 1 ==> 0 0 -1 # 2 ==> ==> F_VECTOR ==> 1 4 3 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 0 ==> {0} # Dimension 1 ==> {1} ==> {2} ==> {3} ==> {0 1} # Dimension 2 ==> {0 2} ==> {0 3} ==> ==> MAXIMAL_CONES ==> {0 1} # Dimension 2 ==> {0 2} ==> {0 3} ==> ==> MULTIPLICITIES ==> 1 # Dimension 2 ==> 1 ==> 1 ==> ring s = 0,x(1..4),wp(1,11,3,19); ideal I = 2*x(1)^2+3*x(1)*x(2)+24*x(3)*x(4), 8*x(1)^3+x(2)*x(3)*x(4)+18*x(3)^2*x(4); fan f = tropicalVariety(I,number(3)); f; ==> _application PolyhedralFan ==> _version 2.2 ==> _type PolyhedralFan ==> ==> AMBIENT_DIM ==> 5 ==> ==> DIM ==> 3 ==> ==> LINEALITY_DIM ==> 1 ==> ==> RAYS ==> -2 -1 1 5 -5 # 0 ==> -1 0 -1 1 0 # 1 ==> -1 0 1 3 -4 # 2 ==> 0 -1 1 1 -1 # 3 ==> 0 0 0 -1 1 # 4 ==> 0 1 -3 1 1 # 5 ==> 0 1 1 5 -7 # 6 ==> ==> N_RAYS ==> 7 ==> ==> 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 7 7 ==> ==> SIMPLICIAL ==> 1 ==> ==> PURE ==> 1 ==> ==> CONES ==> {} # Dimension 1 ==> {0} # Dimension 2 ==> {1} ==> {2} ==> {3} ==> {4} ==> {5} ==> {6} ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {1 4} ==> {2 4} ==> {1 5} ==> {2 6} ==> ==> MAXIMAL_CONES ==> {0 1} # Dimension 3 ==> {0 2} ==> {0 3} ==> {1 4} ==> {2 4} ==> {1 5} ==> {2 6} ==> ==> MULTIPLICITIES ==> 1 # Dimension 3 ==> 1 ==> 1 ==> 1 ==> 1 ==> 1 ==> 1 ==> ```