  
  [1X3. Affine toric varieties[0X
  
  This chapter concerns [5Xtoric[0X commands which deal with the coordinate rings of
  affine toric varieties U_sigma.
  
  
  [1X3.1 Ideals defining affine toric varieties[0X
  
  [1X3.1-1 IdealAffineToricVariety[0X
  
  [2X> IdealAffineToricVariety( [0X[3XL[0X[2X ) _____________________________________[0Xfunction
  
  [13XInput[0X: [3XL[0X is a list generating a cone (as in [10XDualSemigroupGenerators[0X).
  [13XOutput[0X:  the  [5XGAP[0X  ideal  defining  the toric variety associated to the cone
  generated by the vectors in [3XL[0X.
  
  This  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]];[0X                           then
  [10XIdealAffineToricVariety(L);[0X can exhaust GAP's memory allocation.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> J:=IdealAffineToricVariety([[1,0],[3,4]]);[0X
    [4X[ two-sided ideal in PolynomialRing(..., [ x_1, x_2 ]), (3 generators) ][0X
    [4Xgap> GeneratorsOfIdeal(J);[0X
    [4X[ -x_2^2+x_1, -x_2^3+x_1^2, -x_2^4+x_1^3 ][0X
  [4X------------------------------------------------------------------[0X
  
  [1X3.1-2 EmbeddingAffineToricVariety[0X
  
  [2X> EmbeddingAffineToricVariety( [0X[3XL[0X[2X ) _________________________________[0Xfunction
  
  [13XInput[0X: [3XL[0X is a list generating a cone (as in [10XDualSemigroupGenerators[0X).
  [13XOutput[0X:  the toroidal embedding of X=Spec([10XIdealAffineToricVariety(L)[0X) (given
  as a list of multinomials).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> phi:=EmbeddingAffineToricVariety([[1,0],[3,4]]);[0X
    [4X[ x_2, x_1, x_1^2/x_4, x_1^3/x_4^2, x_1^4/x_4^3 ][0X
    [4Xgap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;[0X
    [4Xgap> phi:=EmbeddingAffineToricVariety(L);[0X
    [4X[ x_3, x_2, x_1/x_5, x_1/x_6 ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
