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.
AllCodimensions tels 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.
i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3); |
i2 : time R' = integralClosure(R, Strategy=>{RadicalCodim1})
-- used 0.640168 seconds
o2 = R'
o2 : QuotientRing
|
i3 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3); |
i4 : time R' = integralClosure(R)
-- used 0.507191 seconds
o4 = R'
o4 : QuotientRing
|
i5 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3); |
i6 : time R' = integralClosure(R, Strategy=>{AllCodimensions})
-- used 0.51115 seconds
o6 = R'
o6 : QuotientRing
|
i7 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3); |
i8 : time R' = integralClosure(R, Strategy=>{RadicalCodim1, AllCodimensions})
-- used 0.641896 seconds
o8 = R'
o8 : QuotientRing
|