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 1.85611 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 3 4 4
- 321690417048x - 482535625572x y - 482535625572x y - 402113021310x y + 2653945940646x*y - 965071251144x - 27343685449
o6 = {---------------------------------------------------------------------------------------------------------------------------
3
1688874689502x - 1688874
----------------------------------------------------------------------------------------------------------------------------
3 3 4 3 2 2 3
08x y + 23885513465814x*y + 482535625572y + 1286761668192x + 361580028761952x y - 148299282259128x*y + 4021130213100y -
----------------------------------------------------------------------------------------------------------------------------
2 2 2
689502x y + 442324323441x*y + 80422604262y
----------------------------------------------------------------------------------------------------------------------------
2 3 2
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 1.74756 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 3 4 4
- 321690417048x - 482535625572x y - 482535625572x y - 402113021310x y + 2653945940646x*y - 965071251144x - 2734368544
o12 = {--------------------------------------------------------------------------------------------------------------------------
3
1688874689502x - 168887
---------------------------------------------------------------------------------------------------------------------------
3 3 4 3 2 2 3
908x y + 23885513465814x*y + 482535625572y + 1286761668192x + 361580028761952x y - 148299282259128x*y + 4021130213100y
---------------------------------------------------------------------------------------------------------------------------
2 2 2
4689502x y + 442324323441x*y + 80422604262y
---------------------------------------------------------------------------------------------------------------------------
2 3 2
- 29595518368416y y + 6y - 16y
-------------------, --------------, x, y}
2x - 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 1.89951 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 2.02668 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 1.99587 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 2.07809 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 0.0369188 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 0.0514789 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 0.0883196 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 0.0517311 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 0.0517831 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 2
o60 = ideal (- 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)
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 2 3 2 3
o64 = ideal (a d - b d - b e + c e - d e - d*e , b c - c - a*b*d + c*d + c*d*e, a c - c - a*b*d + c*d - a*b*e + c*d*e, b -
---------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 3 2 2
b*c - a*c*d + b*d*e, a*b - a*c - b*c*d, a b - b*c - a*c*d - a*c*e, a - a*c - 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 0.0738109 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 2
o71 = ideal (- 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)
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 0.11497 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 0.261521 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 .000437162 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .0950026 sec #fractions 6]
[step 1: time .112141 sec #fractions 6]
-- used 0.224705 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 .000428794 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .0940998 sec #fractions 6]
[step 1: time .11633 sec #fractions 6]
-- used 0.213601 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 .000596958 sec #minors 1]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .124535 sec #fractions 6]
[step 1: time .321331 sec #fractions 6]
-- used 0.449226 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 2 2
o96 = ideal (w d + w e - 2a + 2b + d*e + e , w b - 2w a + 2b*d + b*e, w a - 2w b - a*e, 2w + w e - 2a + 2d*e
0,0 0,0 0,0 0,1 0,0 0,1 0,1 0,0
---------------------------------------------------------------------------------------------------------------------------
2 2 2 2
+ e , w w + w e - 2a*b, w - 4b - e )
0,0 0,1 0,1 0,0
o96 : Ideal of QQ[w , w , a, b, d, e]
0,0 0,1
|
The list of strategies may change in the future!