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.