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# binomialUnmixedDecomposition -- Binomial Unmixed Decomposition

## Synopsis

• 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].