systemOfParameters -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal

Synopsis

• Usage:

  J = systemOfParameters I

  J = systemOfParameters (i,I)

• Inputs:
  • I, an ideal, generated by homogeneous elements
  • i, an integer,
• Optional inputs:
  • Verbose => a Boolean value, default value false
  • Attempts => an integer, default value 100
  • Density => a real number, default value 0, or QQ or ZZ; will be converted to RR
  • Seed => ..., default value null
  • Seed => an ideal, default value null
• Outputs:
  • J, an ideal, generated by a homogeneous system of parameters of length i contained in I

Description

i1 : S = ZZ/101[a,b,c,d]

o1 = S

o1 : PolynomialRing

i2 : I = ideal"ab,bc,cd,da"

o2 = ideal (a*b, b*c, c*d, a*d)

o2 : Ideal of S

i3 : codim I

o3 = 2

i4 : setRandomSeed 0

o4 = 0

i5 : inhomogeneousSystemOfParameters I

o5 = | bc+cd bc+ad |

             1       2
o5 : Matrix S  <--- S

i6 : systemOfParameters I

o6 = ideal (a*b, c*d)

o6 : Ideal of S

i7 : systemOfParameters(I, Density => .1, Attempts => 1000, Verbose => true)
Attempts: 1000 Density: .1 Seed: null

o7 = ideal (a*b, c*d)

o7 : Ideal of S

Caveat

See also

• regularSequenceCheck -- how much of a list is regular
• Depth -- Finds the depth of a module or ideal, and systems of parameters in an ideal
• inhomogeneousSystemOfParameters -- generates an inhomogeneous system of parameters