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# binomialUnmixedDecomposition -- Binomial Unmixed Decomposition

- Usage:
    `binomialUnmixedDecomposition I`
- Inputs:
  - I, a binomial ideal
- Optional inputs:
  - Verbose => ..., default value false, generate informative output
- Outputs:
  - a list of unmixed components of I

## Description

This routine returns an unmixed decomposition of a binomial ideal into binomial ideals. The implemented algorithm is a variant of Algorithm 4 in [OS00].

    i1 : R = QQ[x,y,z]

    o1 = R

    o1 : PolynomialRing

    i2 : I = ideal (x^2, x*y, y^2, x*(z^3-1), y*(z^2-1))

                 2        2     3         2
    o2 = ideal (x , x*y, y , x*z  - x, y*z  - y)

    o2 : Ideal of R

    i3 : bud = binomialUnmixedDecomposition I

                  2        2     2         3
    o3 = {ideal (y , x*y, x , y*z  - y, x*z  - x)}

    o3 : List

    i4 : intersect bud == I

    o4 = true

A synonym for this function is BUD.

## Caveat

Apart from unmixedness, properties of the output decomposition are defined only by the course of the algorithm, in particular it is not mesoprimary decomposition of [KM11].

## See also

- BUD -- Binomial Unmixed Decomposition

## Ways to use binomialUnmixedDecomposition:

- `binomialUnmixedDecomposition(Ideal)`

## For the programmer

The object binomialUnmixedDecomposition is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Binomials.m2:1445:0.