genericResidual -- Computes generic residual intersections of an ideal

Usage:

K = genericResidual(s,I)
Inputs:
- s, an integer,
- I, an ideal,
Outputs:
- K, an ideal,

Description

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 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
 -- setting random seed to 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

Ways to use genericResidual:

genericResidual(ZZ,Ideal)

For the programmer

The object genericResidual is a method function.

The source of this document is in ResidualIntersections.m2:886:0.

genericResidual -- Computes generic residual intersections of an ideal

Description

See also

Ways to use genericResidual:

For the programmer