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:
Outputs:
- t, a Boolean value, true if J is a reduction of I, false otherwise

For an ideal I, a subideal J of I is said to be a reduction of I if there exists a nonnegative integer n such that JIⁿ=Iⁿ⁺¹.

This function returns true if J is a reduction of I and returns false if J is not a subideal of I or J is a subideal but not a reduction of I.

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