Overall, the default options are the best. However, sometimes one of these is dramatically better (or worse!). For the examples here, one doesn't notice much difference.
RadicalCodim1 chooses an alternate, often much faster, sometimes much slower, algorithm for computing the radical of ideals. This will often produce a different presentation for the integral closure. Radical chooses yet another such algorithm.
AllCodimensions tells the algorithm to bypass the computation of the S2-ification, but in each iteration of the algorithm, use the radical of the extended Jacobian ideal from the previous step, instead of using only the codimension 1 components of that. This is useful when for some reason the S2-ification is hard to compute, or if the probabilistic algorithm for computing it fails. In general though, this option slows down the computation for many examples.
StartWithOneMinor tells the algorithm to not compute the entire Jacobian ideal, just find one element in it. This is often a bad choice, unless the ideal is large enough that one can't compute the Jacobian ideal. In the future, we plan on using the FastMinors package to compute part of the Jacobian ideal.
SimplifyFractions changes the fractions to hopefully be simpler. Sometimes it succeeds, yet sometimes it makes the fractions worse. This is because of the manner in which fraction fields work. We are hoping that in the future, less drastic change of fractions will happen by default.
Vasconocelos tells the routine to instead of computing Hom(J,J), to instead compute Hom(J^-1, J^-1). This is usually a more time consuming computation, but it does potentially get to the answer in a smaller number of steps.
i1 : S = QQ[x,y,z]
o1 = S
o1 : PolynomialRing
|
i2 : f = ideal (x^8-z^6-y^2*z^4-z^3)
8 2 4 6 3
o2 = ideal(x - y z - z - z )
o2 : Ideal of S
|
i3 : R = S/f
o3 = R
o3 : QuotientRing
|
i4 : time R' = integralClosure R
-- used 0.646738 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 0.494443 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 0.418218 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 0.429441 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 0.580185 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 0.443414 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 0.0359681 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 |
+-----------+
|
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 0.036843 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 0.0511479 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 0.0641067 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 0.0362615 seconds
o56 = R'
o56 : QuotientRing
|
i57 : icFractions R
2
c
o57 = {--, a, b, c, d}
d
o57 : List
|
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 0.088966 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 0.30789 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 0.267151 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 .000361157 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .110628 sec #fractions 6]
[step 1: time .118392 sec #fractions 6]
-- used 0.23208 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 .000459862 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .109942 sec #fractions 6]
[step 1: time .159334 sec #fractions 6]
-- used 0.272373 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
|
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 .000476103 sec #minors 1]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .131675 sec #fractions 6]
[step 1: time .378058 sec #fractions 6]
-- used 0.512628 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
|