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 11.8404 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 12.0368 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 12.6737 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 13.9363 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 13.3935 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 13.7549 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 0.348632 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 0.194358 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 0.697827 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 0.197825 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 0.349477 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 0.498638 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 0.702497 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 1.59774 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 .00120042 sec #minors 4] integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 [step 0: time .60904 sec #fractions 6] [step 1: time .600432 sec #fractions 6] -- used 1.40895 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 .00102739 sec #minors 4] integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 [step 0: time .606434 sec #fractions 6] [step 1: time .850084 sec #fractions 6] -- used 1.65669 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 .00104932 sec #minors 1] integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 [step 0: time .901226 sec #fractions 6] [step 1: time 2.2917 sec #fractions 6] -- used 3.19856 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 |
The list of strategies may change in the future!