returns K = F:J where F is generated by s elements chosend at random from elements of degrees e_1...e_s in the ideal. If the degrees of the the generators of the ideal are d_1<=...<=d_n, then the e_i = d_(n-s+i) if s<=n, and otherwise d_1+1...d_n+1, d_n+1...d_n+1.
The call genericArtinNagata calls genericResidual,and produces a list where the first item is the codepth of (ring I)/K (or -1 if K is not of codim 2), and the second item is K.
i1 : setRandomSeed 0 o1 = 0 |
i2 : S = ZZ/101[a,b,c,d,e] o2 = S o2 : PolynomialRing |
i3 : I = minors(2, random(S^2, S^{3:-1}))
2 2 2
o3 = ideal (45a + 24a*b - 50b + 10a*c + 48b*c + 45c - 49a*d - 50b*d -
------------------------------------------------------------------------
2 2 2 2
10c*d + 23d + 3a*e - 7b*e + 8c*e - 4d*e + 16e , 22a + 45a*b + 17b +
------------------------------------------------------------------------
2 2
36a*c + b*c + 6c - 31a*d - 13b*d - 4c*d + 22d - 27a*e - 30b*e + 44c*e
------------------------------------------------------------------------
2 2 2 2
+ 21d*e + 4e , 24a + 2a*b + 35b + 44a*c + 15b*c + 34c - 41a*d + 18b*d
------------------------------------------------------------------------
2 2
+ 48c*d + 49d + 41a*e - 15b*e + 16c*e - 13d*e - 32e )
o3 : Ideal of S
|
i4 : assert(genericResidual(5,I) == (ideal vars S)^3) |
i5 : (genericArtinNagata(5,I))_0 o5 = 0 |
The object genericResidual is a method function.