# monReduction -- the minimal monomial reduction of a monomial ideal

## Synopsis

• Usage:
monReduction(I)
• Inputs:
• I, ,
• Outputs:
• , 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.