Demontration: Computing with ideals using Singular (early draft)

Status: this sheet is the script of a brief interactive demo during the Material and activities linked to the CRM-2017 school and workshop on “Equivariant Combinatorics”.

Let us define an ideal:

sage: P = QQ['a,b,c,d,e']
sage: P.inject_variables()
Defining a, b, c, d, e

sage: p1 = 3*c^2 - 4*b*d + a*e
sage: p2 = -2*b*c*d + 3*a*d^2 + 3*b^2*e - 4*a*c*e
sage: p3 = 8*b^2*d^2 - 9*a*c*d^2 - 9*b^2*c*e + 9*a*c^2*e + 2*a*b*d*e - a^2*e^2
sage: I = Ideal([p1, p2, p3])

sage: a in I
False
sage: (p1*a - b * p2)  in I
True
sage: I.dimension()
3

The calculations are actually carried out by Singular. Many more advanced features are not directly exposed in Sage, in which case one need to call singular directly. Here we follow the instructions from Singular’s manual to compute a free resolution of this ideal:

sage: res = I._singular_().mres(0); res
[1]:
   _[1]=3*c^2-4*b*d+a*e
   _[2]=2*b*c*d-3*a*d^2-3*b^2*e+4*a*c*e
[2]:
   _[1]=2*b*c*d*gen(1)-3*a*d^2*gen(1)-3*b^2*e*gen(1)+4*a*c*e*gen(1)-3*c^2*gen(2)+4*b*d*gen(2)-a*e*gen(2)
[3]:
   _[1]=0
[4]:
   _[1]=gen(1)
[5]:
   _[1]=0

And its Betti numbers:

sage: res.betti()
1     0     0
0     1     0
0     1     0
0     0     1

TODO: explore how to get the nice pretty printing provided by Singular:

sage: res.betti().print("betti")             # todo: not implemented
  File "<ipython-input-19-cd5e12c00f