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# binomialCellularDecomposition -- Binomial Cellular Decomposition

## Synopsis

• Usage:
binomialCellularDecomposition I
• Inputs:
• I, a binomial ideal
• Optional inputs:
• ReturnCellVars => ..., default value false
• Verbose => ..., default value false, generate informative output
• Outputs:
• a list of cellular ideals whose intersection is I or a list of pairs of these ideals and their cellular variables if the option ReturnCellVars => true is used

## Description

A binomial ideal I is called cellular if modulo I every variable in the polynomial ring is either a non-zerodivisor or nilpotent. This routine returns a minimal cellular decomposition of a binomial ideal.
 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal (x*y-z, x*z-y^2) 2 o2 = ideal (x*y - z, - y + x*z) o2 : Ideal of R i3 : bcd = binomialCellularDecomposition I 2 2 o3 = {ideal (y - x*z, x*y - z, x - y), ideal (z, y)} o3 : List i4 : intersect bcd == I o4 = true i5 : binomialCellularDecomposition (I, ReturnCellVars=>true, Verbose=>false) 2 2 o5 = {{ideal (y - x*z, x*y - z, x - y), {x, y, z}}, {ideal (z, y), {x}}} o5 : List
A synonym for this function is BCD.If the option Verbose is set (default), then output about the number of components found so far will be generated.