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integralClosure(...,Strategy=>...) -- control the algorithm used

Synopsis

Description

Overall, the default options are the best. However, sometimes one of these is dramatically better (or worse!). For the examples here, one doesn't notice much difference.

RadicalCodim1 chooses an alternate, often much faster, sometimes much slower, algorithm for computing the radical of ideals. This will often produce a different presentation for the integral closure. Radical chooses yet another such algorithm.

AllCodimensions tells the algorithm to bypass the computation of the S2-ification, but in each iteration of the algorithm, use the radical of the extended Jacobian ideal from the previous step, instead of using only the codimension 1 components of that. This is useful when for some reason the S2-ification is hard to compute, or if the probabilistic algorithm for computing it fails. In general though, this option slows down the computation for many examples.

StartWithOneMinor tells the algorithm to not compute the entire Jacobian ideal, just find one element in it. This is often a bad choice, unless the ideal is large enough that one can't compute the Jacobian ideal. In the future, we plan on using the FastMinors package to compute part of the Jacobian ideal.

SimplifyFractions changes the fractions to hopefully be simpler. Sometimes it succeeds, yet sometimes it makes the fractions worse. This is because of the manner in which fraction fields work. We are hoping that in the future, less drastic change of fractions will happen by default.

Vasconocelos tells the routine to instead of computing Hom(J,J), to instead compute Hom(J^-1, J^-1). This is usually a more time consuming computation, but it does potentially get to the answer in a smaller number of steps.

i1 : S = QQ[x,y,z]

o1 = S

o1 : PolynomialRing
i2 : f = ideal (x^8-z^6-y^2*z^4-z^3)

            8    2 4    6    3
o2 = ideal(x  - y z  - z  - z )

o2 : Ideal of S
i3 : R = S/f

o3 = R

o3 : QuotientRing
i4 : time R' = integralClosure R
     -- used 0.852181 seconds

o4 = R'

o4 : QuotientRing
i5 : netList (ideal R')_*

     +------------------------------------------------------------------------+
     |         3                                                              |
o5 = |w   z - x                                                               |
     | 4,0                                                                    |
     +------------------------------------------------------------------------+
     |         2 2    4                                                       |
     |w   x - y z  - z  - z                                                   |
     | 1,1                                                                    |
     +------------------------------------------------------------------------+
     |     4                                                                  |
     |w   x  - w   z                                                          |
     | 4,0      1,1                                                           |
     +------------------------------------------------------------------------+
     |            2 2     2 3    2                                            |
     |w   w    - x y z - x z  - x                                             |
     | 4,0 1,1                                                                |
     +------------------------------------------------------------------------+
     | 2           3 2     2      6 2                                         |
     |w   z + w   x y z - w    + x z                                          |
     | 4,0     4,0         1,1                                                |
     +------------------------------------------------------------------------+
     | 2           4 2        2                                               |
     |w   x + w   x y  - w   y z - w                                          |
     | 4,0     4,0        1,1       1,1                                       |
     +------------------------------------------------------------------------+
     | 3      2   3 2        6       4 2       2 4      6       2        3    |
     |w    + w   x y  + w   x z - x*y z  - 2x*y z  - x*z  - 2x*y z - 2x*z  - x|
     | 4,0    4,0        4,0                                                  |
     +------------------------------------------------------------------------+
i6 : icFractions R

       3   2 2    4
      x   y z  + z  + z
o6 = {--, -------------, x, y, z}
       z        x

o6 : List
i7 : S = QQ[x,y,z]

o7 = S

o7 : PolynomialRing
i8 : f = ideal (x^8-z^6-y^2*z^4-z^3)

            8    2 4    6    3
o8 = ideal(x  - y z  - z  - z )

o8 : Ideal of S
i9 : R = S/f

o9 = R

o9 : QuotientRing
i10 : time R' = integralClosure(R, Strategy => Radical)
     -- used 0.84124 seconds

o10 = R'

o10 : QuotientRing
i11 : netList (ideal R')_*

      +------------------------------------------------------------------------+
      |         3                                                              |
o11 = |w   z - x                                                               |
      | 4,0                                                                    |
      +------------------------------------------------------------------------+
      |         2 2    4                                                       |
      |w   x - y z  - z  - z                                                   |
      | 1,1                                                                    |
      +------------------------------------------------------------------------+
      |     4                                                                  |
      |w   x  - w   z                                                          |
      | 4,0      1,1                                                           |
      +------------------------------------------------------------------------+
      |            2 2     2 3    2                                            |
      |w   w    - x y z - x z  - x                                             |
      | 4,0 1,1                                                                |
      +------------------------------------------------------------------------+
      | 2           3 2     2      6 2                                         |
      |w   z + w   x y z - w    + x z                                          |
      | 4,0     4,0         1,1                                                |
      +------------------------------------------------------------------------+
      | 2           4 2        2                                               |
      |w   x + w   x y  - w   y z - w                                          |
      | 4,0     4,0        1,1       1,1                                       |
      +------------------------------------------------------------------------+
      | 3      2   3 2        6       4 2       2 4      6       2        3    |
      |w    + w   x y  + w   x z - x*y z  - 2x*y z  - x*z  - 2x*y z - 2x*z  - x|
      | 4,0    4,0        4,0                                                  |
      +------------------------------------------------------------------------+
i12 : icFractions R

        3   2 2    4
       x   y z  + z  + z
o12 = {--, -------------, x, y, z}
        z        x

o12 : List
i13 : S = QQ[x,y,z]

o13 = S

o13 : PolynomialRing
i14 : f = ideal (x^8-z^6-y^2*z^4-z^3)

             8    2 4    6    3
o14 = ideal(x  - y z  - z  - z )

o14 : Ideal of S
i15 : R = S/f

o15 = R

o15 : QuotientRing
i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
     -- used 0.664522 seconds

o16 = R'

o16 : QuotientRing
i17 : netList (ideal R')_*

      +------------------------------------------------------------------------+
      |         3                                                              |
o17 = |w   z - x                                                               |
      | 4,0                                                                    |
      +------------------------------------------------------------------------+
      |         2 2    4                                                       |
      |w   x - y z  - z  - z                                                   |
      | 1,1                                                                    |
      +------------------------------------------------------------------------+
      |     4                                                                  |
      |w   x  - w   z                                                          |
      | 4,0      1,1                                                           |
      +------------------------------------------------------------------------+
      |            2 2     2 3    2                                            |
      |w   w    - x y z - x z  - x                                             |
      | 4,0 1,1                                                                |
      +------------------------------------------------------------------------+
      | 2           3 2     2      6 2                                         |
      |w   z + w   x y z - w    + x z                                          |
      | 4,0     4,0         1,1                                                |
      +------------------------------------------------------------------------+
      | 2           4 2        2                                               |
      |w   x + w   x y  - w   y z - w                                          |
      | 4,0     4,0        1,1       1,1                                       |
      +------------------------------------------------------------------------+
      | 3      2   3 2        6       4 2       2 4      6       2        3    |
      |w    + w   x y  + w   x z - x*y z  - 2x*y z  - x*z  - 2x*y z - 2x*z  - x|
      | 4,0    4,0        4,0                                                  |
      +------------------------------------------------------------------------+
i18 : S = QQ[x,y,z]

o18 = S

o18 : PolynomialRing
i19 : f = ideal (x^8-z^6-y^2*z^4-z^3)

             8    2 4    6    3
o19 = ideal(x  - y z  - z  - z )

o19 : Ideal of S
i20 : R = S/f

o20 = R

o20 : QuotientRing
i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
     -- used 0.606557 seconds

o21 = R'

o21 : QuotientRing
i22 : netList (ideal R')_*

      +------------------------------------------------------------------------+
      |         3                                                              |
o22 = |w   z - x                                                               |
      | 4,0                                                                    |
      +------------------------------------------------------------------------+
      |         2 2    4                                                       |
      |w   x - y z  - z  - z                                                   |
      | 1,0                                                                    |
      +------------------------------------------------------------------------+
      |            2 2     2 3    2                                            |
      |w   w    - x y z - x z  - x                                             |
      | 4,0 1,0                                                                |
      +------------------------------------------------------------------------+
      |     4                                                                  |
      |w   x  - w   z                                                          |
      | 4,0      1,0                                                           |
      +------------------------------------------------------------------------+
      | 2           3 2     2      6 2                                         |
      |w   z + w   x y z - w    + x z                                          |
      | 4,0     4,0         1,0                                                |
      +------------------------------------------------------------------------+
      | 2           4 2        2                                               |
      |w   x + w   x y  - w   y z - w                                          |
      | 4,0     4,0        1,0       1,0                                       |
      +------------------------------------------------------------------------+
      | 3      2   3 2        6       4 2       2 4      6       2        3    |
      |w    + w   x y  + w   x z - x*y z  - 2x*y z  - x*z  - 2x*y z - 2x*z  - x|
      | 4,0    4,0        4,0                                                  |
      +------------------------------------------------------------------------+
i23 : S = QQ[x,y,z]

o23 = S

o23 : PolynomialRing
i24 : f = ideal (x^8-z^6-y^2*z^4-z^3)

             8    2 4    6    3
o24 = ideal(x  - y z  - z  - z )

o24 : Ideal of S
i25 : R = S/f

o25 = R

o25 : QuotientRing
i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
     -- used 1.17761 seconds

o26 = R'

o26 : QuotientRing
i27 : netList (ideal R')_*

      +------------------------------------------------------------------------+
      |         3                                                              |
o27 = |w   z - x                                                               |
      | 4,0                                                                    |
      +------------------------------------------------------------------------+
      |         2 2    4                                                       |
      |w   x - y z  - z  - z                                                   |
      | 1,1                                                                    |
      +------------------------------------------------------------------------+
      |     4                                                                  |
      |w   x  - w   z                                                          |
      | 4,0      1,1                                                           |
      +------------------------------------------------------------------------+
      |            2 2     2 3    2                                            |
      |w   w    - x y z - x z  - x                                             |
      | 4,0 1,1                                                                |
      +------------------------------------------------------------------------+
      | 2           3 2     2      6 2                                         |
      |w   z + w   x y z - w    + x z                                          |
      | 4,0     4,0         1,1                                                |
      +------------------------------------------------------------------------+
      | 2           4 2        2                                               |
      |w   x + w   x y  - w   y z - w                                          |
      | 4,0     4,0        1,1       1,1                                       |
      +------------------------------------------------------------------------+
      | 3      2   3 2        6       4 2       2 4      6       2        3    |
      |w    + w   x y  + w   x z - x*y z  - 2x*y z  - x*z  - 2x*y z - 2x*z  - x|
      | 4,0    4,0        4,0                                                  |
      +------------------------------------------------------------------------+
i28 : S = QQ[x,y,z]

o28 = S

o28 : PolynomialRing
i29 : f = ideal (x^8-z^6-y^2*z^4-z^3)

             8    2 4    6    3
o29 = ideal(x  - y z  - z  - z )

o29 : Ideal of S
i30 : R = S/f

o30 = R

o30 : QuotientRing
i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
     -- used 0.811243 seconds

o31 = R'

o31 : QuotientRing
i32 : netList (ideal R')_*

      +------------------------------------------------------------------------+
      |         3                                                              |
o32 = |w   z - x                                                               |
      | 4,0                                                                    |
      +------------------------------------------------------------------------+
      |         2 2    4                                                       |
      |w   x - y z  - z  - z                                                   |
      | 1,1                                                                    |
      +------------------------------------------------------------------------+
      |     4                                                                  |
      |w   x  - w   z                                                          |
      | 4,0      1,1                                                           |
      +------------------------------------------------------------------------+
      |            2 2     2 3    2                                            |
      |w   w    - x y z - x z  - x                                             |
      | 4,0 1,1                                                                |
      +------------------------------------------------------------------------+
      | 2           3 2     2      6 2                                         |
      |w   z + w   x y z - w    + x z                                          |
      | 4,0     4,0         1,1                                                |
      +------------------------------------------------------------------------+
      | 2           4 2        2                                               |
      |w   x + w   x y  - w   y z - w                                          |
      | 4,0     4,0        1,1       1,1                                       |
      +------------------------------------------------------------------------+
      | 3      2   3 2        6       4 2       2 4      6       2        3    |
      |w    + w   x y  + w   x z - x*y z  - 2x*y z  - x*z  - 2x*y z - 2x*z  - x|
      | 4,0    4,0        4,0                                                  |
      +------------------------------------------------------------------------+
i33 : S = QQ[a,b,c,d]

o33 = S

o33 : PolynomialRing
i34 : f = monomialCurveIdeal(S,{1,3,4})

                         3      2     2    2    3    2
o34 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o34 : Ideal of S
i35 : R = S/f

o35 = R

o35 : QuotientRing
i36 : time R' = integralClosure R
     -- used 0.0978372 seconds

o36 = R'

o36 : QuotientRing
i37 : netList (ideal R')_*

      +-----------+
o37 = |b*c - a*d  |
      +-----------+
      |         2 |
      |w   d - c  |
      | 0,0       |
      +-----------+
      |w   c - b*d|
      | 0,0       |
      +-----------+
      |w   b - a*c|
      | 0,0       |
      +-----------+
      |         2 |
      |w   a - b  |
      | 0,0       |
      +-----------+
      | 2         |
      |w    - a*d |
      | 0,0       |
      +-----------+

Rational Quartic

i38 : S = QQ[a,b,c,d]

o38 = S

o38 : PolynomialRing
i39 : I = monomialCurveIdeal(S,{1,3,4})

                         3      2     2    2    3    2
o39 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o39 : Ideal of S
i40 : R = S/I

o40 = R

o40 : QuotientRing
i41 : time R' = integralClosure(R, Strategy => Radical)
     -- used 0.101284 seconds

o41 = R'

o41 : QuotientRing
i42 : icFractions R

        2
       c
o42 = {--, a, b, c, d}
        d

o42 : List
i43 : S = QQ[a,b,c,d]

o43 = S

o43 : PolynomialRing
i44 : I = monomialCurveIdeal(S,{1,3,4})

                         3      2     2    2    3    2
o44 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o44 : Ideal of S
i45 : R = S/I

o45 = R

o45 : QuotientRing
i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
     -- used 0.149661 seconds

o46 = R'

o46 : QuotientRing
i47 : icFractions R

       b*d
o47 = {---, a, b, c, d}
        c

o47 : List
i48 : S = QQ[a,b,c,d]

o48 = S

o48 : PolynomialRing
i49 : I = monomialCurveIdeal(S,{1,3,4})

                         3      2     2    2    3    2
o49 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o49 : Ideal of S
i50 : R = S/I

o50 = R

o50 : QuotientRing
i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
     -- used 0.0545561 seconds

o51 = R'

o51 : QuotientRing
i52 : icFractions R

        2
       c
o52 = {--, a, b, c, d}
        d

o52 : List
i53 : S = QQ[a,b,c,d]

o53 = S

o53 : PolynomialRing
i54 : I = monomialCurveIdeal(S,{1,3,4})

                         3      2     2    2    3    2
o54 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o54 : Ideal of S
i55 : R = S/I

o55 = R

o55 : QuotientRing
i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
     -- used 0.0603635 seconds

o56 = R'

o56 : QuotientRing
i57 : icFractions R

        2
       c
o57 = {--, a, b, c, d}
        d

o57 : List

Projected Veronese

i58 : S' = QQ[symbol a .. symbol f]

o58 = S'

o58 : PolynomialRing
i59 : M' = genericSymmetricMatrix(S',a,3)

o59 = | a b c |
      | b d e |
      | c e f |

               3        3
o59 : Matrix S'  <--- S'
i60 : I' = minors(2,M')

                2                                                  2         
o60 = ideal (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, -
      -----------------------------------------------------------------------
                                              2
      c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)

o60 : Ideal of S'
i61 : center = ideal(b,c,e,a-d,d-f)

o61 = ideal (b, c, e, a - d, d - f)

o61 : Ideal of S'
i62 : S = QQ[a,b,c,d,e]

o62 = S

o62 : PolynomialRing
i63 : p = map(S'/I',S,gens center)

                                                                   S'
o63 = map (------------------------------------------------------------------------------------------------------------------, S, {b, c, e, a - d, d - f})
               2                                                  2                                                  2
           (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, - c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)

                                                                      S'
o63 : RingMap ------------------------------------------------------------------------------------------------------------------ <--- S
                  2                                                  2                                                  2
              (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, - c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)
i64 : I = kernel p

              2     2     2     2     2       2   2     3              2  
o64 = ideal (a d - b d - b e + c e - d e - d*e , b c - c  - a*b*d + c*d  +
      -----------------------------------------------------------------------
              2     3              2                   3      2          
      c*d*e, a c - c  - a*b*d + c*d  - a*b*e + c*d*e, b  - b*c  - a*c*d +
      -----------------------------------------------------------------------
                2      2           2       2                   3      2  
      b*d*e, a*b  - a*c  - b*c*d, a b - b*c  - a*c*d - a*c*e, a  - a*c  -
      -----------------------------------------------------------------------
                                 2
      b*c*d - b*c*e - a*d*e - a*e )

o64 : Ideal of S
i65 : betti res I

             0 1  2 3 4
o65 = total: 1 7 10 5 1
          0: 1 .  . . .
          1: . .  . . .
          2: . 7 10 5 1

o65 : BettiTally
i66 : R = S/I

o66 = R

o66 : QuotientRing
i67 : time R' = integralClosure(R, Strategy => Radical)
     -- used 0.0899121 seconds

o67 = R'

o67 : QuotientRing
i68 : icFractions R

        2    2
       b  - c
o68 = {-------, a, b, c, d, e}
          d

o68 : List
i69 : S' = QQ[a..f]

o69 = S'

o69 : PolynomialRing
i70 : M' = genericSymmetricMatrix(S',a,3)

o70 = | a b c |
      | b d e |
      | c e f |

               3        3
o70 : Matrix S'  <--- S'
i71 : I' = minors(2,M')

                2                                                  2         
o71 = ideal (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, -
      -----------------------------------------------------------------------
                                              2
      c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)

o71 : Ideal of S'
i72 : center = ideal(b,e,a-d,d-f)

o72 = ideal (b, e, a - d, d - f)

o72 : Ideal of S'
i73 : S = QQ[a,b,d,e]

o73 = S

o73 : PolynomialRing
i74 : p = map(S'/I',S,gens center)

                                                                   S'
o74 = map (------------------------------------------------------------------------------------------------------------------, S, {b, e, a - d, d - f})
               2                                                  2                                                  2
           (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, - c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)

                                                                      S'
o74 : RingMap ------------------------------------------------------------------------------------------------------------------ <--- S
                  2                                                  2                                                  2
              (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, - c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)
i75 : I = kernel p

             4     2 2    4    2 2    2       2       2 2
o75 = ideal(a  - 2a b  + b  - b d  - a d*e - b d*e - a e )

o75 : Ideal of S
i76 : betti res I

             0 1
o76 = total: 1 1
          0: 1 .
          1: . .
          2: . .
          3: . 1

o76 : BettiTally
i77 : R = S/I

o77 = R

o77 : QuotientRing
i78 : time R' = integralClosure(R, Strategy => Radical)
     -- used 0.504602 seconds

o78 = R'

o78 : QuotientRing
i79 : icFractions R

        2    2   2     3      2
       a  - b   a b - b  + b*d  + b*d*e
o79 = {-------, -----------------------, a, b, d, e}
        d + e          a*d + a*e

o79 : List
i80 : S = QQ[a,b,d,e]

o80 = S

o80 : PolynomialRing
i81 : R = S/sub(I,S)

o81 = R

o81 : QuotientRing
i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
     -- used 0.47419 seconds

o82 = R'

o82 : QuotientRing
i83 : icFractions R

        2    2   2     3      2
       a  - b   a b - b  + b*d  + b*d*e
o83 = {-------, -----------------------, a, b, d, e}
        d + e          a*d + a*e

o83 : List
i84 : S = QQ[a,b,d,e]

o84 = S

o84 : PolynomialRing
i85 : R = S/sub(I,S)

o85 = R

o85 : QuotientRing
i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
 [jacobian time .000485355 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .213785 sec  #fractions 6]
 [step 1:   time .278674 sec  #fractions 6]
     -- used 0.497376 seconds

o86 = R'

o86 : QuotientRing
i87 : icFractions R

        2    2   2     3      2
       a  - b   a b - b  + b*d  + b*d*e
o87 = {-------, -----------------------, a, b, d, e}
        d + e          a*d + a*e

o87 : List
i88 : S = QQ[a,b,d,e]

o88 = S

o88 : PolynomialRing
i89 : R = S/sub(I,S)

o89 = R

o89 : QuotientRing
i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
 [jacobian time .000927707 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .253217 sec  #fractions 6]
 [step 1:   time .233537 sec  #fractions 6]
     -- used 0.495152 seconds

o90 = R'

o90 : QuotientRing
i91 : icFractions R

        2    2   2     3      2
       a  - b   a b - b  + b*d  + b*d*e
o91 = {-------, -----------------------, a, b, d, e}
        d + e          a*d + a*e

o91 : List

One can give several of these options together. Although note that only one of AllCodimensions, RadicalCodim1, Radical will be used.

i92 : S = QQ[a,b,d,e]

o92 = S

o92 : PolynomialRing
i93 : R = S/sub(I,S)

o93 = R

o93 : QuotientRing
i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
 [jacobian time .000657092 sec #minors 1]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .158294 sec  #fractions 6]
 [step 1:   time .480489 sec  #fractions 6]
     -- used 0.642842 seconds

o94 = R'

o94 : QuotientRing
i95 : icFractions R

         2     2          2   2     3      2
       2a  - 2b  - d*e - e   a b - b  + b*d  + b*d*e
o95 = {--------------------, -----------------------, a, b, d, e}
               d + e                a*d + a*e

o95 : List
i96 : ideal R'

                               2     2          2                         
o96 = ideal (w   d + w   e - 2a  + 2b  + d*e + e , w   b - 2w   a + 2b*d +
              0,0     0,0                           0,0      0,1          
      -----------------------------------------------------------------------
                                   2               2           2            
      b*e, w   a - 2w   b - a*e, 2w    + w   e - 2a  + 2d*e + e , w   w    +
            0,0      0,1           0,1    0,0                      0,0 0,1  
      -----------------------------------------------------------------------
                     2       2    2
      w   e - 2a*b, w    - 4b  - e )
       0,1           0,0

o96 : Ideal of QQ[w   ..w   , a..b, d..e]
                   0,0   0,1

Further information

Caveat

The list of strategies may change in the future!

Functions with optional argument named Strategy :