# findGensUpToIrrelevance -- creates a list of subsets of the minimal generators that generate a given ideal up to saturation

## Synopsis

• Usage:
findGensUpToIrrelevance(n,I,irr)
findGensUpToIrrelevance(n,I,X)
• Inputs:
• I, an ideal
• n, an integer, size of subset of minimal generators of I that may generate I up to saturation with irr
• irr, an ideal, irrelevant ideal
• X, , normal toric variety whose Cox ring contains I
• Outputs:
• a list, all subsets of size n of generators of I that generate I up to saturation with irr

## Description

Given an ideal I, integer n, and irrelevant ideal irr, findGensUpToIrrelevance searches through all n-subsets of the generators of I. If a subset generates the same irr-saturated ideal as the irr-saturation of I, then that subset is added to a list. After running through all subsets, the list is returned.

 `i1 : R = ZZ/101[x_0,x_1,x_2,x_3,x_4,Degrees=>{2:{1,0},3:{0,1}}];` ```i2 : B = intersect(ideal(x_0,x_1),ideal(x_2,x_3,x_4)); o2 : Ideal of R``` ```i3 : I = ideal(x_0^2*x_2^2+x_1^2*x_3^2+x_0*x_1*x_4^2, x_0^3*x_4+x_1^3*(x_2+x_3)); o3 : Ideal of R``` ```i4 : findGensUpToIrrelevance(2,I,B) o4 = {{0, 1}} o4 : List```

## Caveat

If no subset of generators generates the ideal up to saturation, then the empty list is returned.

## Ways to use findGensUpToIrrelevance :

• findGensUpToIrrelevance(ZZ,Ideal,Ideal)
• findGensUpToIrrelevance(ZZ,Ideal,NormalToricVariety)