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toricIndispensableSet -- the indispensable set of toric binomials

Description

A binomial $x^u - x^v$ of a toric ideal $I_A$ is called indispensable if it belongs to every minimal Markov basis of A. The set of all indispensable elements is called the indispensable set, and is often denoted with $S(A)$.

This method computes the indispensable set of a matrix $A$ and returns the indispensibles as the rows of a matrix. Similarly to markovBases, if a ring $R$ is supplied, then the result is an ideal generated by the indispensable set.

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

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

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

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

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

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

If the optional argument ReturnFiberValues is set to true, then the function instead returns the list of fibers of the indispensable binomials.

i4 : toricIndispensableSet(A, ReturnFiberValues => true)

o4 = {{18}, {16}, {17}, {29}}

o4 : List

The function computes the indispensable elements by checking the connected components of the fiber graph of $A$; see fiberGraph. A fiber gives rise to an indispensable element if and only if it has exactly two connected components and each component contains a single a single point in the fiber.

See also

Ways to use toricIndispensableSet:

  • toricIndispensableSet(Matrix)
  • toricIndispensableSet(Matrix,Ring)
  • toricIndispensableSet(Matrix,Matrix) (missing documentation)

For the programmer

The object toricIndispensableSet is a method function with options.


The source of this document is in AllMarkovBases.m2:1024:0.