  
  [1X3 [33X[0;0YAffine toric varieties[133X[101X
  
  [33X[0;0YThis chapter concerns [5Xtoric[105X commands which deal with the coordinate rings of
  affine toric varieties [22XU_σ[122X.[133X
  
  
  [1X3.1 [33X[0;0YIdeals defining affine toric varieties[133X[101X
  
  [1X3.1-1 IdealAffineToricVariety[101X
  
  [29X[2XIdealAffineToricVariety[102X( [3XL[103X ) [32X function
  
  [33X[0;0Y[13XInput[113X: [3XL[103X is a list generating a cone (as in [10XDualSemigroupGenerators[110X).[133X
  [33X[0;0Y[13XOutput[113X:  the  [5XGAP[105X  ideal  defining  the toric variety associated to the cone
  generated by the vectors in [3XL[103X.[133X
  
  [33X[0;0YThis  computation  is  not  very efficient and should not be used for ideals
  with      many      generators.     For     example,     if     you     take
  [3XL:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]];[103X                           then
  [10XIdealAffineToricVariety(L);[110X can exhaust GAP's memory allocation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ:=IdealAffineToricVariety([[0,1],[2,-1]]);[127X[104X
    [4X[28X<two-sided ideal in Rationals[x_1,x_2,x_3], (1 generators)>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfIdeal(J);[127X[104X
    [4X[28X[ -x_2^2+x_1 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.1-2 EmbeddingAffineToricVariety[101X
  
  [29X[2XEmbeddingAffineToricVariety[102X( [3XL[103X ) [32X function
  
  [33X[0;0Y[13XInput[113X: [3XL[103X is a list generating a cone (as in [10XDualSemigroupGenerators[110X).[133X
  [33X[0;0Y[13XOutput[113X:  the toroidal embedding of [22XX=Spec[122X([10XIdealAffineToricVariety(L)[110X) (given
  as a list of multinomials).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xphi:=EmbeddingAffineToricVariety([[1,0],[3,4]]);[127X[104X
    [4X[28X[ x_2, x_1, x_1^2/x_4, x_1^3/x_4^2, x_1^4/x_4^3 ][128X[104X
    [4X[25Xgap>[125X [27XL:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;[127X[104X
    [4X[25Xgap>[125X [27Xphi:=EmbeddingAffineToricVariety(L);[127X[104X
    [4X[28X[ x_3, x_2, x_1/x_5, x_1/x_6 ][128X[104X
  [4X[32X[104X
  
