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markovBases -- all minimal Markov bases of a configuration matrix

Description

This method produces all minimal Markov bases of a given configuration matrix $A \in \ZZ^{d \times n}$. By default, the output is formatted in the same way as toricMarkov: each Markov basis is a $k \times n$ matrix whose rows correspond to the elements of the Markov basis.

i1 : netList markovBases matrix "3,4,5" -- unique Markov basis

     +-----------+
o1 = || 1 -2 1  ||
     || 3 -1 -1 ||
     || 2 1  -2 ||
     +-----------+
i2 : netList markovBases matrix "1,2,3"

     +-----------+
o2 = || 2 -1 0  ||
     || 1 1  -1 ||
     +-----------+
     || 2 -1 0  ||
     || 3 0  -1 ||
     +-----------+
i3 : netList markovBases matrix "1,2,3,4"

     +--------------+
o3 = || 2 -1 0  0  ||
     || 1 1  -1 0  ||
     || 0 2  0  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 1 1  -1 0  ||
     || 1 0  1  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 1 1  -1 0  ||
     || 2 1  0  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 1 1  -1 0  ||
     || 4 0  0  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 3 0  -1 0  ||
     || 0 2  0  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 3 0  -1 0  ||
     || 1 0  1  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 3 0  -1 0  ||
     || 2 1  0  -1 ||
     +--------------+
     || 2 -1 0  0  ||
     || 3 0  -1 0  ||
     || 4 0  0  -1 ||
     +--------------+
i4 : netList markovBases matrix "1,2,3,4;4,5,6,7"

     +-------------+
o4 = || 1 -2 1  0 ||
     || 1 -1 -1 1 ||
     || 0 1  -2 1 ||
     +-------------+

Similarly to toricMarkov, we may also specify a ring $R$. In this case, the method produces a list of ideals in $R$ with each ideal generated by a different minimal Markov basis of $A$.

i5 : markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])

              2                           2        3
o5 = {ideal (x  - x , x x  - x ), ideal (x  - x , x  - x )}
              1    2   1 2    3           1    2   1    3

o5 : List
i6 : gens \ (markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3]))

o6 = {| x_1^2-x_2 x_1x_2-x_3 |, | x_1^2-x_2 x_1^3-x_3 |}

o6 : List

See also

Ways to use markovBases:

  • markovBases(Matrix)
  • markovBases(Matrix,Ring)

For the programmer

The object markovBases is a method function.


The source of this document is in AllMarkovBases.m2:871:0.