Computing symbolic powers of an ideal

Given an ideal, symbolicPower computes a given symbolic power.

i1 : B = QQ[x,y,z];

i2 : I = ideal(x*(y^3-z^3),y*(z^3-y^3),z*(x^3-y^3));

o2 : Ideal of B

i3 : J = symbolicPower(I,3)

             3 6     9      3 3 4     6 4    3 7    3 7   6 3 2    9 2    6 5
o3 = ideal (x y z - y z - 2x y z  + 2y z  + x z  - y z , x y z  - y z  - x z 
     ------------------------------------------------------------------------
         3 3 5     6 5     3 8     3 8   9 3    9 3     6 6     6 6     3 9  
     - 2x y z  + 3y z  + 2x z  - 2y z , x z  - y z  - 3x z  + 3y z  + 3x z  -
     ------------------------------------------------------------------------
       3 9   12     9 3     6 6    3 9     11       8 3       5 6      2 9 
     3y z , y   - 3y z  + 3y z  - y z , x*y   - 3x*y z  + 3x*y z  - x*y z ,
     ------------------------------------------------------------------------
      2 10     2 7 3     2 4 6    2   9   3 9     9 3     3 3 6     6 6  
     x y   - 3x y z  + 3x y z  - x y*z , x y  - 3y z  - 3x y z  + 6y z  +
     ------------------------------------------------------------------------
       3 9     3 9
     2x z  - 3y z )

o3 : Ideal of B

Various algorithms are used, in the following order:

1. If $I$ is squarefree monomial ideal, intersects the powers of the associated primes of $I$;

2. If $I$ is monomial ideal, but not squarefree, takes an irredundant primary decomposition of $I$ and intersects the powers of those ideals;

3. If $I$ is a saturated homogeneous ideal in a polynomial ring whose height is one less than the dimension of the ring, returns the saturation of $I^n$;

4. If all the associated primes of $I$ have the same height, computes a primary decomposition of $I^n$ and intersects the components with radical $I$;

5. If all else fails, compares the radicals of a primary decomposition of $I^n$ with the associated primes of $I$, and intersects the components corresponding to minimal primes.