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IntegralClosure :: integralClosure(...,Strategy=>...)

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 FastLinAlg 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 0.396835 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 0.38259 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 0.393313 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 0.457188 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 0.549562 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 0.441227 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 0.0357252 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 0.0639939 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 0.0530945 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 0.0658259 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 0.0358447 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 0.0891766 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 0.305431 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 0.224633 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 .000327542 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .109439 sec  #fractions 6]
 [step 1:   time .115816 sec  #fractions 6]
     -- used 0.228083 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 .000318756 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .102293 sec  #fractions 6]
 [step 1:   time .145853 sec  #fractions 6]
     -- used 0.250964 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 .000478315 sec #minors 1]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .160586 sec  #fractions 6]
 [step 1:   time .359215 sec  #fractions 6]
     -- used 0.522771 seconds

o94 = R'

o94 : QuotientRing
i95 : icFractions R

         2     2    2            2   2     3      2
       2a  - 2b  - d  - 3d*e - 2e   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            2                  
o96 = ideal (w   d + w   e - 2a  + 2b  + d  + 3d*e + 2e , w   b - 2w   a +
              0,0     0,0                                  0,0      0,1   
      -----------------------------------------------------------------------
                                           2               2            2 
      3b*d + 2b*e, w   a - 2w   b + a*d, 2w    + w   e - 2a  + 3d*e + 2e ,
                    0,0      0,1           0,1    0,0                     
      -----------------------------------------------------------------------
                                         2       2     2    2            2
      w   w    + w   d + 2w   e - 2a*b, w    + 4a  - 8b  - d  - 4d*e - 4e )
       0,0 0,1    0,1      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 :