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# toricIndispensableSet -- the indispensable set of toric binomials

- Usage:
    `S = toricIndispensableSet A`
    `I = toricIndispensableSet(A, R)`
- Inputs:
  - A, a matrix, the configuration matrix
  - R, a ring, with one generator for each column of $A$
- Optional inputs:
  - ReturnFiberValues => a Boolean value, default value false, whether to return the fibers of indispensable binomials
- Outputs:
  - S, a matrix, indispensable set of toric binomials
  - I, an ideal, an ideal in $R$ generated by the indispensable 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

- markovBases -- all minimal Markov bases of a configuration matrix
- fiberGraph -- generating fibers of a configuration matrix

## 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AllMarkovBases.m2:1024:0.