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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]

o1 = S

o1 : PolynomialRing
i2 : f = ideal (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)

              7    6     5      3 2    4
o2 = ideal(- x  + x  - 4x y - 2x y  + y )

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

o3 = R

o3 : QuotientRing
i4 : time R' = integralClosure R
     -- used 3.31276 seconds

o4 = R'

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

     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |                  3     2                                                                                                                                                                                                                                         |
o5 = |2w   x - w   y - y  - 6y  + 16y                                                                                                                                                                                                                                   |
     |  2,0     2,0                                                                                                                                                                                                                                                     |
     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |      2                   2                     2                           2     3                2                                                                                                                                                              |
     |4w   x  - 4w   x*y + w   y  - 14w   x*y + 6w   y  - 52w   x + 18w   y - 8x*y  + 4y  - 864x*y + 380y  - 416y                                                                                                                                                       |
     |  4,0       4,0       4,0        2,0        2,0        2,0       2,0                                                                                                                                                                                              |
     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |                  2                      2           2               3       2                                                                                                                                                                                    |
     |- 2w   x*y + w   y  + 4w   x - 2w   y + w    + 7w   y  + 12w   y + 4y  + 424y  - 864y                                                                                                                                                                             |
     |    4,0       4,0       4,0      4,0     2,0     2,0        2,0                                                                                                                                                                                                   |
     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |     2          2      4      3       2 2        3      4      3      2         2      3      2                                                                                                                                                                   |
     |w   y  + 21w   y  + 16x  + 24x y + 24x y  + 20x*y  + 15y  - 16x  + 40x y + 72x*y  + 76y  + 32y                                                                                                                                                                    |
     | 4,0        2,0                                                                                                                                                                                                                                                   |
     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     | 2                                        2                                             2                             4       3        2 2        3       4        3        2           2       3       2                    2                                    |
     |w    + 48w   w    + 1204w   x*y - 602w   y  - 2168w   x + 1204w   y + 32w    - 3738w   y  - 3392w   y + 896w    - 560x  - 624x y - 380x y  - 88x*y  + 126y  + 1888x  + 1312x y + 1912x*y  + 584y  - 944x  + 3408x*y - 247532y  - 640x + 468032y + 256             |
     | 4,0      4,0 2,0        4,0          4,0          4,0         4,0       4,0        2,0          2,0        2,0                                                                                                                                                   |
     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |                     2          3                      2           3           2                 5       4        3 2       2 3         4       5        4        3         2 2          3        4      3       2          2         3       2                  2|
     |w   w   y + 98w   x*y  - 49w   y  - 84w   x*y + 30w   y  - 112w   y  - 384w   y  + 32w   y + 896x  + 896x y + 680x y  + 464x y  + 300x*y  + 188y  - 1088x  + 2448x y + 2712x y  + 2136x*y  + 1300y  + 64x  - 784x y + 704x*y  - 20176y  + 128x  - 256x*y + 17056y |
     | 4,0 2,0       4,0          4,0        4,0         4,0         2,0         2,0        2,0                                                                                                                                                                         |
     +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
i6 : icFractions R

                     5                4                 3 2                2 
      - 321690417048x  - 482535625572x y - 482535625572x y  - 402113021310x y
o6 = {-----------------------------------------------------------------------
                                                                             
                                                                             
     ------------------------------------------------------------------------
     3                   4                4                 3                
       + 2653945940646x*y  - 965071251144x  - 2734368544908x y + 238855134658
     ------------------------------------------------------------------------
                                              3                 2            
                                1688874689502x  - 1688874689502x y + 44232432
     ------------------------------------------------------------------------
          3                4                 3                   2           
     14x*y  + 482535625572y  + 1286761668192x  + 361580028761952x y - 1482992
     ------------------------------------------------------------------------
            2               2                                                
     3441x*y  + 80422604262y
     ------------------------------------------------------------------------
                2                 3                  2   3     2
     82259128x*y  + 4021130213100y  - 29595518368416y   y  + 6y  - 16y
     -------------------------------------------------, --------------, x, y}
                                                            2x - y

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

o7 = S

o7 : PolynomialRing
i8 : f = ideal (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)

              7    6     5      3 2    4
o8 = ideal(- x  + x  - 4x y - 2x y  + y )

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

o9 = R

o9 : QuotientRing
i10 : time R' = integralClosure(R, Strategy => Radical)
     -- used 2.83318 seconds

o10 = R'

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

      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  3     2                                                                                                                                                                                                                                         |
o11 = |2w   x - w   y - y  - 6y  + 16y                                                                                                                                                                                                                                   |
      |  2,0     2,0                                                                                                                                                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2                   2                     2                           2     3                2                                                                                                                                                              |
      |4w   x  - 4w   x*y + w   y  - 14w   x*y + 6w   y  - 52w   x + 18w   y - 8x*y  + 4y  - 864x*y + 380y  - 416y                                                                                                                                                       |
      |  4,0       4,0       4,0        2,0        2,0        2,0       2,0                                                                                                                                                                                              |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  2                      2           2               3       2                                                                                                                                                                                    |
      |- 2w   x*y + w   y  + 4w   x - 2w   y + w    + 7w   y  + 12w   y + 4y  + 424y  - 864y                                                                                                                                                                             |
      |    4,0       4,0       4,0      4,0     2,0     2,0        2,0                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |     2          2      4      3       2 2        3      4      3      2         2      3      2                                                                                                                                                                   |
      |w   y  + 21w   y  + 16x  + 24x y + 24x y  + 20x*y  + 15y  - 16x  + 40x y + 72x*y  + 76y  + 32y                                                                                                                                                                    |
      | 4,0        2,0                                                                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      | 2                                        2                                             2                             4       3        2 2        3       4        3        2           2       3       2                    2                                    |
      |w    + 48w   w    + 1204w   x*y - 602w   y  - 2168w   x + 1204w   y + 32w    - 3738w   y  - 3392w   y + 896w    - 560x  - 624x y - 380x y  - 88x*y  + 126y  + 1888x  + 1312x y + 1912x*y  + 584y  - 944x  + 3408x*y - 247532y  - 640x + 468032y + 256             |
      | 4,0      4,0 2,0        4,0          4,0          4,0         4,0       4,0        2,0          2,0        2,0                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                     2          3                      2           3           2                 5       4        3 2       2 3         4       5        4        3         2 2          3        4      3       2          2         3       2                  2|
      |w   w   y + 98w   x*y  - 49w   y  - 84w   x*y + 30w   y  - 112w   y  - 384w   y  + 32w   y + 896x  + 896x y + 680x y  + 464x y  + 300x*y  + 188y  - 1088x  + 2448x y + 2712x y  + 2136x*y  + 1300y  + 64x  - 784x y + 704x*y  - 20176y  + 128x  - 256x*y + 17056y |
      | 4,0 2,0       4,0          4,0        4,0         4,0         2,0         2,0        2,0                                                                                                                                                                         |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
i12 : icFractions R

                      5                4                 3 2                2
       - 321690417048x  - 482535625572x y - 482535625572x y  - 402113021310x 
o12 = {----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
       3                   4                4                 3              
      y  + 2653945940646x*y  - 965071251144x  - 2734368544908x y + 2388551346
      -----------------------------------------------------------------------
                                                3                 2          
                                  1688874689502x  - 1688874689502x y + 442324
      -----------------------------------------------------------------------
             3                4                 3                   2        
      5814x*y  + 482535625572y  + 1286761668192x  + 361580028761952x y - 1482
      -----------------------------------------------------------------------
               2               2                                             
      323441x*y  + 80422604262y
      -----------------------------------------------------------------------
                    2                 3                  2   3     2
      99282259128x*y  + 4021130213100y  - 29595518368416y   y  + 6y  - 16y
      ----------------------------------------------------, --------------,
                                                                2x - y
      -----------------------------------------------------------------------
      x, y}

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

o13 = S

o13 : PolynomialRing
i14 : f = ideal (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)

               7    6     5      3 2    4
o14 = ideal(- x  + x  - 4x y - 2x y  + y )

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

o15 = R

o15 : QuotientRing
i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
     -- used 2.86464 seconds

o16 = R'

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

      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  3     2                                                                                                                                                                                                                                         |
o17 = |2w   x - w   y - y  - 6y  + 16y                                                                                                                                                                                                                                   |
      |  2,0     2,0                                                                                                                                                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2                   2                     2                           2     3                2                                                                                                                                                              |
      |4w   x  - 4w   x*y + w   y  - 14w   x*y + 6w   y  - 52w   x + 18w   y - 8x*y  + 4y  - 864x*y + 380y  - 416y                                                                                                                                                       |
      |  4,0       4,0       4,0        2,0        2,0        2,0       2,0                                                                                                                                                                                              |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  2                      2           2               3       2                                                                                                                                                                                    |
      |- 2w   x*y + w   y  + 4w   x - 2w   y + w    + 7w   y  + 12w   y + 4y  + 424y  - 864y                                                                                                                                                                             |
      |    4,0       4,0       4,0      4,0     2,0     2,0        2,0                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |     2          2      4      3       2 2        3      4      3      2         2      3      2                                                                                                                                                                   |
      |w   y  + 21w   y  + 16x  + 24x y + 24x y  + 20x*y  + 15y  - 16x  + 40x y + 72x*y  + 76y  + 32y                                                                                                                                                                    |
      | 4,0        2,0                                                                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      | 2                                        2                                             2                             4       3        2 2        3       4        3        2           2       3       2                    2                                    |
      |w    + 48w   w    + 1204w   x*y - 602w   y  - 2168w   x + 1204w   y + 32w    - 3738w   y  - 3392w   y + 896w    - 560x  - 624x y - 380x y  - 88x*y  + 126y  + 1888x  + 1312x y + 1912x*y  + 584y  - 944x  + 3408x*y - 247532y  - 640x + 468032y + 256             |
      | 4,0      4,0 2,0        4,0          4,0          4,0         4,0       4,0        2,0          2,0        2,0                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                     2          3                      2           3           2                 5       4        3 2       2 3         4       5        4        3         2 2          3        4      3       2          2         3       2                  2|
      |w   w   y + 98w   x*y  - 49w   y  - 84w   x*y + 30w   y  - 112w   y  - 384w   y  + 32w   y + 896x  + 896x y + 680x y  + 464x y  + 300x*y  + 188y  - 1088x  + 2448x y + 2712x y  + 2136x*y  + 1300y  + 64x  - 784x y + 704x*y  - 20176y  + 128x  - 256x*y + 17056y |
      | 4,0 2,0       4,0          4,0        4,0         4,0         2,0         2,0        2,0                                                                                                                                                                         |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
i18 : S = QQ[x,y]

o18 = S

o18 : PolynomialRing
i19 : f = ideal (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)

               7    6     5      3 2    4
o19 = ideal(- x  + x  - 4x y - 2x y  + y )

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

o20 = R

o20 : QuotientRing
i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
     -- used 3.16381 seconds

o21 = R'

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

      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  3     2                                                                                                                                                                                                                                         |
o22 = |2w   x - w   y - y  - 6y  + 16y                                                                                                                                                                                                                                   |
      |  2,0     2,0                                                                                                                                                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2                   2                     2                           2     3                2                                                                                                                                                              |
      |4w   x  - 4w   x*y + w   y  - 14w   x*y + 6w   y  - 52w   x + 18w   y - 8x*y  + 4y  - 864x*y + 380y  - 416y                                                                                                                                                       |
      |  4,0       4,0       4,0        2,0        2,0        2,0       2,0                                                                                                                                                                                              |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  2                      2           2               3       2                                                                                                                                                                                    |
      |- 2w   x*y + w   y  + 4w   x - 2w   y + w    + 7w   y  + 12w   y + 4y  + 424y  - 864y                                                                                                                                                                             |
      |    4,0       4,0       4,0      4,0     2,0     2,0        2,0                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |     2          2      4      3       2 2        3      4      3      2         2      3      2                                                                                                                                                                   |
      |w   y  + 21w   y  + 16x  + 24x y + 24x y  + 20x*y  + 15y  - 16x  + 40x y + 72x*y  + 76y  + 32y                                                                                                                                                                    |
      | 4,0        2,0                                                                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      | 2                                        2                                             2                             4       3        2 2        3       4        3        2           2       3       2                    2                                    |
      |w    + 48w   w    + 1204w   x*y - 602w   y  - 2168w   x + 1204w   y + 32w    - 3738w   y  - 3392w   y + 896w    - 560x  - 624x y - 380x y  - 88x*y  + 126y  + 1888x  + 1312x y + 1912x*y  + 584y  - 944x  + 3408x*y - 247532y  - 640x + 468032y + 256             |
      | 4,0      4,0 2,0        4,0          4,0          4,0         4,0       4,0        2,0          2,0        2,0                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                     2          3                      2           3           2                 5       4        3 2       2 3         4       5        4        3         2 2          3        4      3       2          2         3       2                  2|
      |w   w   y + 98w   x*y  - 49w   y  - 84w   x*y + 30w   y  - 112w   y  - 384w   y  + 32w   y + 896x  + 896x y + 680x y  + 464x y  + 300x*y  + 188y  - 1088x  + 2448x y + 2712x y  + 2136x*y  + 1300y  + 64x  - 784x y + 704x*y  - 20176y  + 128x  - 256x*y + 17056y |
      | 4,0 2,0       4,0          4,0        4,0         4,0         2,0         2,0        2,0                                                                                                                                                                         |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
i23 : S = QQ[x,y]

o23 = S

o23 : PolynomialRing
i24 : f = ideal (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)

               7    6     5      3 2    4
o24 = ideal(- x  + x  - 4x y - 2x y  + y )

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

o25 = R

o25 : QuotientRing
i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
     -- used 3.19934 seconds

o26 = R'

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

      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  3     2                                                                                                                                                                                                                                         |
o27 = |2w   x - w   y - y  - 6y  + 16y                                                                                                                                                                                                                                   |
      |  2,0     2,0                                                                                                                                                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2                   2                     2                           2     3                2                                                                                                                                                              |
      |4w   x  - 4w   x*y + w   y  - 14w   x*y + 6w   y  - 52w   x + 18w   y - 8x*y  + 4y  - 864x*y + 380y  - 416y                                                                                                                                                       |
      |  4,0       4,0       4,0        2,0        2,0        2,0       2,0                                                                                                                                                                                              |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  2                      2           2               3       2                                                                                                                                                                                    |
      |- 2w   x*y + w   y  + 4w   x - 2w   y + w    + 7w   y  + 12w   y + 4y  + 424y  - 864y                                                                                                                                                                             |
      |    4,0       4,0       4,0      4,0     2,0     2,0        2,0                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |     2          2      4      3       2 2        3      4      3      2         2      3      2                                                                                                                                                                   |
      |w   y  + 21w   y  + 16x  + 24x y + 24x y  + 20x*y  + 15y  - 16x  + 40x y + 72x*y  + 76y  + 32y                                                                                                                                                                    |
      | 4,0        2,0                                                                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      | 2                                        2                                             2                             4       3        2 2        3       4        3        2           2       3       2                    2                                    |
      |w    + 48w   w    + 1204w   x*y - 602w   y  - 2168w   x + 1204w   y + 32w    - 3738w   y  - 3392w   y + 896w    - 560x  - 624x y - 380x y  - 88x*y  + 126y  + 1888x  + 1312x y + 1912x*y  + 584y  - 944x  + 3408x*y - 247532y  - 640x + 468032y + 256             |
      | 4,0      4,0 2,0        4,0          4,0          4,0         4,0       4,0        2,0          2,0        2,0                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                     2          3                      2           3           2                 5       4        3 2       2 3         4       5        4        3         2 2          3        4      3       2          2         3       2                  2|
      |w   w   y + 98w   x*y  - 49w   y  - 84w   x*y + 30w   y  - 112w   y  - 384w   y  + 32w   y + 896x  + 896x y + 680x y  + 464x y  + 300x*y  + 188y  - 1088x  + 2448x y + 2712x y  + 2136x*y  + 1300y  + 64x  - 784x y + 704x*y  - 20176y  + 128x  - 256x*y + 17056y |
      | 4,0 2,0       4,0          4,0        4,0         4,0         2,0         2,0        2,0                                                                                                                                                                         |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
i28 : S = QQ[x,y]

o28 = S

o28 : PolynomialRing
i29 : f = ideal (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)

               7    6     5      3 2    4
o29 = ideal(- x  + x  - 4x y - 2x y  + y )

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

o30 = R

o30 : QuotientRing
i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
     -- used 3.17563 seconds

o31 = R'

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

      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  3     2                                                                                                                                                                                                                                         |
o32 = |2w   x - w   y - y  - 6y  + 16y                                                                                                                                                                                                                                   |
      |  2,0     2,0                                                                                                                                                                                                                                                     |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |      2                   2                     2                           2     3                2                                                                                                                                                              |
      |4w   x  - 4w   x*y + w   y  - 14w   x*y + 6w   y  - 52w   x + 18w   y - 8x*y  + 4y  - 864x*y + 380y  - 416y                                                                                                                                                       |
      |  4,0       4,0       4,0        2,0        2,0        2,0       2,0                                                                                                                                                                                              |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                  2                      2           2               3       2                                                                                                                                                                                    |
      |- 2w   x*y + w   y  + 4w   x - 2w   y + w    + 7w   y  + 12w   y + 4y  + 424y  - 864y                                                                                                                                                                             |
      |    4,0       4,0       4,0      4,0     2,0     2,0        2,0                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |     2          2      4      3       2 2        3      4      3      2         2      3      2                                                                                                                                                                   |
      |w   y  + 21w   y  + 16x  + 24x y + 24x y  + 20x*y  + 15y  - 16x  + 40x y + 72x*y  + 76y  + 32y                                                                                                                                                                    |
      | 4,0        2,0                                                                                                                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      | 2                                        2                                             2                             4       3        2 2        3       4        3        2           2       3       2                    2                                    |
      |w    + 48w   w    + 1204w   x*y - 602w   y  - 2168w   x + 1204w   y + 32w    - 3738w   y  - 3392w   y + 896w    - 560x  - 624x y - 380x y  - 88x*y  + 126y  + 1888x  + 1312x y + 1912x*y  + 584y  - 944x  + 3408x*y - 247532y  - 640x + 468032y + 256             |
      | 4,0      4,0 2,0        4,0          4,0          4,0         4,0       4,0        2,0          2,0        2,0                                                                                                                                                   |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                     2          3                      2           3           2                 5       4        3 2       2 3         4       5        4        3         2 2          3        4      3       2          2         3       2                  2|
      |w   w   y + 98w   x*y  - 49w   y  - 84w   x*y + 30w   y  - 112w   y  - 384w   y  + 32w   y + 896x  + 896x y + 680x y  + 464x y  + 300x*y  + 188y  - 1088x  + 2448x y + 2712x y  + 2136x*y  + 1300y  + 64x  - 784x y + 704x*y  - 20176y  + 128x  - 256x*y + 17056y |
      | 4,0 2,0       4,0          4,0        4,0         4,0         2,0         2,0        2,0                                                                                                                                                                         |
      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
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 0.0386716 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 0.077898 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 0.118126 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 0.0814977 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 0.0400459 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 0.113308 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 0.199369 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 0.404921 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 .000461099 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .136192 sec  #fractions 6]
 [step 1:   time .175851 sec  #fractions 6]
     -- used 0.315701 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 .000458901 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .173215 sec  #fractions 6]
 [step 1:   time .141397 sec  #fractions 6]
     -- used 0.318326 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 .000591853 sec #minors 1]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2

 [step 0:   time .193779 sec  #fractions 6]
 [step 1:   time .489888 sec  #fractions 6]
     -- used 0.687247 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!