Overall, the default options are the best. However, sometimes one of these is dramatically better (or worse!). For the examples here, one doesn't notice much difference.
RadicalCodim1 chooses an alternate, often much faster, sometimes much slower, algorithm for computing the radical of ideals. This will often produce a different presentation for the integral closure. Radical chooses yet another such algorithm.
AllCodimensions tells the algorithm to bypass the computation of the S2-ification, but in each iteration of the algorithm, use the radical of the extended Jacobian ideal from the previous step, instead of using only the codimension 1 components of that. This is useful when for some reason the S2-ification is hard to compute, or if the probabilistic algorithm for computing it fails. In general though, this option slows down the computation for many examples.
StartWithOneMinor tells the algorithm to not compute the entire Jacobian ideal, just find one element in it. This is often a bad choice, unless the ideal is large enough that one can't compute the Jacobian ideal. In the future, we plan on using the FastLinAlg package to compute part of the Jacobian ideal.
SimplifyFractions changes the fractions to hopefully be simpler. Sometimes it succeeds, yet sometimes it makes the fractions worse. This is because of the manner in which fraction fields work. We are hoping that in the future, less drastic change of fractions will happen by default.
Vasconocelos tells the routine to instead of computing Hom(J,J), to instead compute Hom(J^-1, J^-1). This is usually a more time consuming computation, but it does potentially get to the answer in a smaller number of steps.
i1 : S = QQ[x,y,z]
o1 = S
o1 : PolynomialRing
|
i2 : f = ideal (x^8-z^6-y^2*z^4-z^3)
8 2 4 6 3
o2 = ideal(x - y z - z - z )
o2 : Ideal of S
|
i3 : R = S/f
o3 = R
o3 : QuotientRing
|
i4 : time R' = integralClosure R
-- used 0.408475 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 0.405249 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 0.407158 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 0.441367 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 0.582843 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 0.423145 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 0.0408791 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 0.0416434 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 0.0651878 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 0.0528497 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 0.0410399 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 0.0846578 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 0.301811 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 0.274148 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 .000422235 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .109986 sec #fractions 6]
[step 1: time .119137 sec #fractions 6]
-- used 0.232591 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 .000438706 sec #minors 4]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .107848 sec #fractions 6]
[step 1: time .198189 sec #fractions 6]
-- used 0.309527 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 .000588532 sec #minors 1]
integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
[step 0: time .128984 sec #fractions 6]
[step 1: time .376245 sec #fractions 6]
-- used 0.508751 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
|