hilbertBasis(Matrix) -- calculates the Hilbert basis of the cone; invokes "hilbert" from 4ti2

• Function: hilbertBasis

• Usage:

  hilbertBasis(A) or hilbertBasis(A, InputType => "lattice")

• Inputs:
  • A, a matrix, defining the cone $\{z : Az = 0, z \ge 0 \}$
• Optional inputs:
  • InputType => a string, default value null, use the string "lattice" if rows of A specify a lattice basis
  • Precision => an integer or a string, default value 32, 32, 64, or "gmp", the precision of the integers used during the computation
• Outputs:
  • B, a matrix, whose rows form the Hilbert basis of the cone $\{z : Az = 0, z \ge 0 \}$ or the cone $\{z A : z \text{ is an integral vector and } z A \ge 0 \}$ if InputType => "lattice" is used

i1 : A = matrix "1,1,1,1; 1,2,3,4"

o1 = | 1 1 1 1 |
     | 1 2 3 4 |

              2       4
o1 : Matrix ZZ  <-- ZZ

i2 : B = syz A

o2 = | 1  2  |
     | -2 -3 |
     | 1  0  |
     | 0  1  |

              4       2
o2 : Matrix ZZ  <-- ZZ

i3 : hilbertBasis(transpose B)

o3 = | 3 2 1 0 |
     | 0 1 2 3 |
     | 1 1 1 1 |

              3       4
o3 : Matrix ZZ  <-- ZZ

i4 : hilbertBasis(A, InputType => "lattice")

o4 = | 3 2 1 0 |
     | 0 1 2 3 |
     | 1 1 1 1 |

              3       4
o4 : Matrix ZZ  <-- ZZ