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gfanDoesIdealContain -- check ideal membership by the division algorithm



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) + 0, (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 :

For the programmer

The object gfanDoesIdealContain is a method function with options.