getMaxIdeal -- computes a maximal ideal containing a given ideal in a polynomial ring

Synopsis

Usage:

M = getMaxIdeal I

M = getMaxIdeal(I,L)
Inputs:
- I, an ideal, an ideal of a polynomial ring over QQ, ZZ, or ZZ/p for p a prime integer.
- L, a list, a subset of the variables of the ring. This list must contain the variables that appear in the ideal I. By default, L is assumed to be the list of variables in the ring.
Optional inputs:
- Verbose (missing documentation) => an integer, default value 0, which controls the level of output of the method (0, 1, 2, 3, or 4).
Outputs:
- M, an ideal, maximal with respect to the variables in L.

Description

In absence of an input list, getMaxIdeal yields a maximal ideal containing the input ideal I.

i1 : R = ZZ/3[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x*(x-1)*(x-2)*y*(y-1)*(y-2)+1)

            3 3    3       3
o2 = ideal(x y  - x y - x*y  + x*y + 1)

o2 : Ideal of R

i3 : J = getMaxIdeal I

                    2
o3 = ideal (x - y, y  + 1)

o3 : Ideal of R

i4 : isSubset(I,J)

o4 = true

The function isSubset shows that I is contained in our new ideal. To see that J is indeed maximal, consider the codimension and the minimal primes.

i5 : codim J

o5 = 2

i6 : P = minimalPrimes J

                     2
o6 = {ideal (x - y, y  + 1)}

o6 : List

i7 : J == P_0

o7 = true

The optional list argument allows us to restrict our maximal ideal to a polynomial ring defined by a subset of the variables of the ambient ring. Note that the list must contain the variables that appear in the generators of I.

i8 : R = ZZ[x,y,z,a,b,c]

o8 = R

o8 : PolynomialRing

i9 : I = ideal(27,x^2+1)

                 2
o9 = ideal (27, x  + 1)

o9 : Ideal of R

i10 : J = getMaxIdeal(I,{x,y,z})

                    2
o10 = ideal (z, y, x  + 1, 3)

o10 : Ideal of R

i11 : isSubset(I,J)

o11 = true

Ways to use getMaxIdeal :

getMaxIdeal(Ideal)
getMaxIdeal(Ideal,List)

For the programmer

The object getMaxIdeal is a method function with options.