monReduction -- the minimal monomial reduction of a monomial ideal

Synopsis

Usage:

monReduction(I)
Inputs:
- I, a monomial ideal,
Outputs:
- a monomial ideal, the minimal monomial reduction of I

Description

Given a monomial ideal I, this function computes a monomial reduction of I (i.e. a reduction of I which is a monomial ideal), which is inclusion-wise minimal among all monomial reductions of I.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal"x2,xy,y3"

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

o2 : Ideal of R

i3 : J = monReduction I

                     2        3
o3 = monomialIdeal (x , x*y, y )

o3 : MonomialIdeal of R

i4 : J == I

o4 = true

i5 : K = minimalReduction I

            9 3   9 2   1     3 3   1 2
o5 = ideal (-y  + -x  + -x*y, -y  + -x  + x*y)
            4     2     2     4     2

o5 : Ideal of R

i6 : degree J, degree K

o6 = (4, 6)

o6 : Sequence

This function works by finding the extremal rays of NP(I), which correspond to the minimal generators of the monomial reduction of I.

Caveat

As seen above, a monomial minimal reduction need not be a minimal reduction.

Ways to use monReduction :

monReduction(Ideal)

For the programmer