independentSets -- some size-maximal independent subsets of variables modulo an ideal

Synopsis

Usage:
independentSets J
Inputs:
- J, an ideal, or a monomial ideal
Outputs:
- a list, of products of variables. The support of any one of these products is a maximal independent subset of variables modulo J
Optional inputs:
- Limit => an integer, default value infinity, the maximum number of independent sets to be found

Description

An independent set of variables of an ideal J in a polynomial ring R is a set of variables that are algebraically independent modulo J (i.e. there is no polynomial in J involving only these sets of variables.

If the Krull dimension of R/J is d, then a maximal independent set is an independent set having size d.

i1 : R = QQ[a..h];

i2 : I = minors(2,genericMatrix(R,a,2,4))

o2 = ideal (- b*c + a*d, - b*e + a*f, - d*e + c*f, - b*g + a*h, - d*g + c*h,
     ------------------------------------------------------------------------
     - f*g + e*h)

o2 : Ideal of R

i3 : inI = ideal leadTerm I

o3 = ideal (f*g, d*g, b*g, d*e, b*e, b*c)

o3 : Ideal of R

i4 : independentSets I

o4 = {a*b*d*f*h, a*c*d*f*h, a*c*e*f*h, a*c*e*g*h}

o4 : List

i5 : independentSets inI

o5 = {a*b*d*f*h, a*c*d*f*h, a*c*e*f*h, a*c*e*g*h}

o5 : List

The independent sets returned correspond one for one with the minimal primes of smallest codimension of the ideal of lead terms of J.

i6 : I = ideal"abc,bcd,cde,adf,cgh,b3f,a3g"

                                                3    3
o6 = ideal (a*b*c, b*c*d, c*d*e, a*d*f, c*g*h, b f, a g)

o6 : Ideal of R

i7 : minimalPrimes I

o7 = {ideal (c, g, f), ideal (a, d, f, h), ideal (a, b, h, e), ideal (a, b,
     ------------------------------------------------------------------------
     g, e), ideal (a, b, c), ideal (a, b, d, h), ideal (a, c, f), ideal (a,
     ------------------------------------------------------------------------
     d, g, f), ideal (b, g, f, e), ideal (b, d, g)}

o7 : List

i8 : independentSets I

o8 = {a*b*d*e*h, a*c*e*f*h, b*d*e*g*h, d*e*f*g*h}

o8 : List

The optional Limit argument is useful if you need only one, or several such independent sets.

i9 : L = independentSets(I, Limit=>1)

o9 = {a*b*d*e*h}

o9 : List

Often, you want the list of the variables in a maximal independent set, or the list of those not in the set.

i10 : support L_0

o10 = {a, b, d, e, h}

o10 : List

i11 : rsort toList(set gens R - set support L_0)

o11 = {c, f, g}

o11 : List

This function is useful as a subroutine to primary decomposition algorithms.

Ways to use `independentSets` :

independentSets(Ideal)
independentSets(MonomialIdeal)

independentSets -- some size-maximal independent subsets of variables modulo an ideal

Synopsis

Description

See also

Ways to use independentSets :

Ways to use `independentSets` :