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# gfanDoesIdealContain -- check ideal membership by the division algorithm

## Synopsis

• Usage:
B = gfanDoesIdealContain(M,K)
B = gfanDoesIdealContain(L,K)
B = gfanDoesIdealContain(I,K)
• Inputs:
• M, , a marked Groebner basis.
• L, a list, a list of Groebner basis.
• I, an ideal, an ideal with Groebner basis as generator.
• K, a list, a list of polynomials
• Outputs:
• B, , true if every polynomial in K belongs to the ideal generated by L

## Description

This method determines if a list of polynomials is contained in an ideal. The first input has to be in marked polynomial, in order to do this easily one could use gfanBuchberger as shown in the example as gfanBuchberger outputs a Groebner basis in as a markedPolynomialList. Macaulay 2 provides this functionality in the isSubset method.

 i1 : QQ[x,y,z]; i2 : gfanDoesIdealContain(markedPolynomialList{{y*z,x*y,x*z}, {y*z, x*y - y, x*z + z}}, {x*y*z}) o2 = true i3 : L=gfanBuchberger{x*y-y, x*z+z} o3 = {(y*z), (x*z) + z, (x*y) - y} o3 : MarkedPolynomialList i4 : gfanDoesIdealContain(L,{x*y*z}) o4 = true i5 : isSubset(ideal(y*z), ideal(x*y - y, x*z +z)) o5 = true

gfan Documentation

This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
Options:
--remainder:
Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
--multiplier:
Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.


## Ways to use gfanDoesIdealContain :

• gfanDoesIdealContain(Ideal,List)
• gfanDoesIdealContain(List,List)
• gfanDoesIdealContain(MarkedPolynomialList,List)

## For the programmer

The object gfanDoesIdealContain is .