integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain

Synopsis

Function: integralClosure
Usage:

integralClosure J

integralClosure(J, d)

integralClosure(J, f)

integralClosure(J, f, d)
Inputs:
- J, an ideal,
- f, a ring element, optional, an element of J which is a nonzerodivisor in the ring of J. If not give, the first generator of J is used
- d, an integer, optional, default value 1
Optional inputs:
- Keep => a list, default value null, unused
- Limit => an integer, default value infinity, unused
- Variable => a symbol, default value "w", symbol used for new variables
- Verbosity => an integer, default value 0, display a certain amount of detail about the computation
- Strategy => a list, default value {}, of some of the symbols: AllCodimensions, SimplifyFractions, Radical, RadicalCodim1, Vasconcelos. These are passed on to the computation of the integral closure of the Rees algebra of J
Outputs:
- an ideal, the integral closure of $J^d$

Description

The method used is described in Vasconcelos' book, Computational methods in commutative algebra and algebraic geometry, Springer, section 6.6. Basically, one first computes the integral closure of the Rees Algebra of the ideal, and then one reads off the integral closure of any of the powers of the ideal, using linear algebra.

i1 : S = ZZ/32003[a,b,c];

i2 : F = a^2*b^2*c+a^3+b^3+c^3

      2 2     3    3    3
o2 = a b c + a  + b  + c

o2 : S

i3 : J = ideal jacobian ideal F

                2      2    2        2   2 2     2
o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )

o3 : Ideal of S

i4 : time integralClosure J
     -- used 0.776778 seconds

             2 2              2 2                2          2   2     
o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
     ------------------------------------------------------------------------
           2   3               2 2     2   5
     16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)

o4 : Ideal of S

i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
     -- used 0.59032 seconds

             2 2              2 2                2          2   2     
o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
     ------------------------------------------------------------------------
           2   3               2 2     2   5
     16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)

o5 : Ideal of S

i6 : J2' = integralClosure(J,2)

             5      2 4           3       3   2 3      2 4         4       3 
o6 = ideal (b c - 2b c  - 16000a*b  - 3a*c , a b c - 2a c  - 16000b  - 3b*c ,
     ------------------------------------------------------------------------
      3 2     5          4           3   5     2 3          3          4 
     a b c - b c - 16000a  + 16000a*b , a c - a b c - 16000a b + 16000b ,
     ------------------------------------------------------------------------
          6         2 2 2         3          3         4   2 2 4         3 3
     a*b*c  - 16000a b c  - 12000a c - 12000b c - 8003c , a b c  - 16000a c 
     ------------------------------------------------------------------------
             3 3            2   2 3 3         3   2         4 2          2  
     - 16000b c  + 8003a*b*c , a b c  - 16000a b*c  - 16000b c  + 8003a*b c,
     ------------------------------------------------------------------------
      3 2 3         4 2           3 2        2      4   3     2 2 2  
     a b c  - 16000a c  - 16000a*b c  + 8003a b*c, a b*c  + 3a b c  +
     ------------------------------------------------------------------------
          3    2 4 2     3 2         4   3 3 2         4              4   
     8003b c, a b c  + 3a b c + 8003a , a b c  - 16000a b*c - 16000a*b c +
     ------------------------------------------------------------------------
          2 2   4 2 2     2 3         4   5   2     3 2           3   6 2  
     8003a b , a b c  + 3a b c + 8003b , a b*c  + 3a b c + 8003a*b , a c  +
     ------------------------------------------------------------------------
       4           2 2   3 4          4 2       2 3         2 2   4 3   
     3a b*c + 8003a b , a b c - 16000a b  + 3a*b c  - 15997a c , a b c -
     ------------------------------------------------------------------------
           2 4     2   3         2 2   5 2          3 3     3 3  
     16000a b  + 3a b*c  - 15997b c , a b c - 16000a b  + 3a c  -
     ------------------------------------------------------------------------
               2   4 4     2 2 2     4   7      2 2 2         3          3   
     15997a*b*c , a b  + 6a b c  + 9c , a b + 6a b c  + 15997a c - 15997b c +
     ------------------------------------------------------------------------
       4   8            4 2           3 2       5         2      7 3  
     9c , a b*c + 15988a c  + 15997a*b c  + 9a*c  + 15988a b*c, a b  +
     ------------------------------------------------------------------------
           5         4        3           3   10 2      4 2           5  
     15988b c + 8021a  - 27a*b  + 15988a*c , a  b  + 27a b  + 15988a*b  +
     ------------------------------------------------------------------------
          2 3        2 2
     27a*b c  - 7940a c )

o6 : Ideal of S

Sometimes it is useful to give the specific nonzerodivisor $f$ in the ideal.

i7 : assert(integralClosure(J, J_2, 2) == J2')

Caveat

It is usually much faster to use integralClosure(J,d) rather than integralClosure(J^d). Also, the element f (or the first generator of J, if f is not given) must be a nonzero divisor in the ring. This is not checked.

Ways to use this method: