minimalBettiNumbers -- return the minimal Betti numbers of the quotient ring corresponding to a given graded ideal

• Usage:

  minimalBettiNumbers I

• Inputs:
  • I, a graded ideal of a polynomial ring
• Outputs:
  • a Betti tally, the Betti table of the quotient S/I computed using its minimal generators

i1 : S=QQ[x_1..x_4]

o1 = S

o1 : PolynomialRing

i2 : I=ideal(x_1*x_2,x_1*x_3,x_2*x_3)

o2 = ideal (x x , x x , x x )
             1 2   1 3   2 3

o2 : Ideal of S

i3 : J=ideal(join(flatten entries gens I,{x_1*x_2*x_3}))

o3 = ideal (x x , x x , x x , x x x )
             1 2   1 3   2 3   1 2 3

o3 : Ideal of S

i4 : I==J

o4 = true

i5 : betti I==betti J

o5 = false

i6 : minimalBettiNumbers I==minimalBettiNumbers J

o6 = true