isReduction -- Determine whether an ideal is a reduction

Synopsis

• Usage:

  t=isReduction(I,J)

  t=isReduction(I,J,f)

• Inputs:
  • I, an ideal,
  • J, an ideal,
  • f, a ring element, an optional element, which is a non-zerodivisor modulo M and the ring of M
• Optional inputs:
  • BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
  • DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
  • MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
  • PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
  • Strategy => ..., default value null, Choose a strategy for the saturation step
  • Variable => ..., default value w, Choose name for variables in the created ring
• Outputs:
  • t, a Boolean value, true if J is a reduction of I, false otherwise

Description

i1 : S = ZZ/5[x,y]

o1 = S

o1 : PolynomialRing

i2 : I = ideal(x^3,x*y,y^4)

             3        4
o2 = ideal (x , x*y, y )

o2 : Ideal of S

i3 : J = ideal(x*y, x^3+y^4)

                  4    3
o3 = ideal (x*y, y  + x )

o3 : Ideal of S

i4 : isReduction(I,J)

o4 = true

i5 : isReduction(J,I)

o5 = false

i6 : isReduction(I,I)

o6 = true

i7 : g = I_0

      3
o7 = x

o7 : S

i8 : isReduction(I,J,g)

o8 = true

i9 : isReduction(J,I,g)

o9 = false

i10 : isReduction(I,I,g)

o10 = true

See also

• minimalReduction -- Find a minimal reduction of an ideal
• reductionNumber -- Reduction number of one ideal with respect to another