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Tropical :: tropicalVariety

tropicalVariety -- the tropical variety associated to an ideal

Synopsis

Description

This method takes an ideal and computes the tropical variety associated to it. By default the ideal is assumed to be prime. If this is not the case the default answer will not necessarily give the correct answer. In this case use the optional argument Prime=>false. By default the tropicalVariety command computes multiplicities but setting computeMultiplicities=>false turns this off. This only saves time if Prime is set to false. The ideal I is not assumed to be homogeneous. The optional argument IsHomogeneous=>true allows the user to assert that the ideal is homogeneous.

i1 : QQ[x,y];
i2 : I=ideal(x+y+1);

o2 : Ideal of QQ[x, y]
i3 : T=tropicalVariety(I);
i4 : rays(T)

o4 = | 1 -1 0 |
     | 0 -1 1 |

              2        3
o4 : Matrix ZZ  <--- ZZ
i5 : maxCones(T)

o5 = {{1}, {0}, {2}}

o5 : List
i6 : linealitySpace T

o6 = 0

              2
o6 : Matrix ZZ  <--- 0
i7 : fVector fan T

o7 = {1, 3}

o7 : List
i8 : multiplicities(T)

o8 = {1, 1, 1}

o8 : List
i9 : QQ[x,y,z,w];
i10 : I=intersect(ideal(x+y+z+w),ideal(x-y,y-z));

o10 : Ideal of QQ[x, y, z, w]
i11 : T= tropicalVariety(I,Prime=>false);
i12 : rays(T)

o12 = | 1  3  1  -1 1  -1 -1 -1 |
      | 1  -1 1  3  -1 1  -1 -1 |
      | 1  -1 -1 -1 1  1  3  -1 |
      | -3 -1 -1 -1 -1 -1 -1 3  |

               4        8
o12 : Matrix ZZ  <--- ZZ
i13 : maxCones(T)

o13 = {{0}, {1, 2}, {1, 4}, {1, 7}, {2, 3}, {4, 6}, {3, 5}, {3, 7}, {5, 6},
      -----------------------------------------------------------------------
      {6, 7}}

o13 : List
i14 : multiplicities(T)

o14 = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

o14 : List
i15 : linealitySpace T

o15 = | 1 |
      | 1 |
      | 1 |
      | 1 |

               4        1
o15 : Matrix ZZ  <--- ZZ
i16 : QQ[x,y,z,w];
i17 : I=intersect(ideal(x+y+z+1),ideal(x^2-y*z));

o17 : Ideal of QQ[x, y, z, w]
i18 : T= tropicalVariety(I,Prime=>false,ComputeMultiplicities=>false);
i19 : rays(T)

o19 = | -4 4 0 0 4 4 |
      | -4 0 4 0 8 0 |
      | -4 0 0 4 0 8 |
      | -3 1 1 1 3 3 |

               4        6
o19 : Matrix ZZ  <--- ZZ
i20 : maxCones(T)

o20 = {{0, 1}, {0, 2}, {0, 3}, {0, 4}, {0, 5}, {2, 3}, {1, 4}, {1, 5}, {2,
      -----------------------------------------------------------------------
      4}, {3, 5}, {4, 5}}

o20 : List
i21 : linealitySpace T

o21 = | 0 |
      | 0 |
      | 0 |
      | 1 |

               4        1
o21 : Matrix ZZ  <--- ZZ
i22 : multiplicities(T)

o22 = {{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}}

o22 : List

Ways to use tropicalVariety :