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toricUniversalMarkov -- the universal Markov basis

Description

The universal Markov basis, often denoted $U(A)$, of a configuration matrix $A$ is the union of all minimal Markov bases of $A$. This method computes the universal Markov basis of $A$ and returns the elements as the rows of a matrix. Similarly to markovBases, if a ring $R$ is supplied, then the result is an ideal generated by the universal Markov basis.

i1 : A = matrix "7,8,9,10";

              1       4
o1 : Matrix ZZ  <-- ZZ
i2 : toricUniversalMarkov A

o2 = | -1 2  -1 0  |
     | -1 1  1  -1 |
     | 0  -1 2  -1 |
     | 4  0  -2 -1 |
     | 4  -1 0  -2 |
     | 3  1  -1 -2 |
     | 3  0  1  -3 |
     | 2  2  0  -3 |

              8       4
o2 : Matrix ZZ  <-- ZZ
i3 : toricUniversalMarkov(A, QQ[x_1 .. x_4])

             2                       2          4    2     4      2   3    
o3 = ideal (x  - x x , x x  - x x , x  - x x , x  - x x , x  - x x , x x  -
             2    1 3   2 3    1 4   3    2 4   1    3 4   1    2 4   1 2  
     ------------------------------------------------------------------------
        2   3      3   2 2    3
     x x , x x  - x , x x  - x )
      3 4   1 3    4   1 2    4

o3 : Ideal of QQ[x ..x ]
                  1   4

The function computes the universal Markov basis elements using the the fiber graph of $A$; see fiberGraph. An irreducible binomial $x^u - x^v$ is an element of the universal Markov basis if and only if $u$ and $v$ belong to different connected components of the fiber graph.

See also

Ways to use toricUniversalMarkov:

  • toricUniversalMarkov(Matrix)
  • toricUniversalMarkov(Matrix,Ring)

For the programmer

The object toricUniversalMarkov is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/AllMarkovBases.m2:1068:0.