multiplierIdeal(MonomialIdeal,QQ) -- multiplier ideal

multiplier ideal of a monomial ideal

• Usage:

  multiplierIdeal(I,t)

• Inputs:
  • I, a monomial ideal, a monomial ideal in a polynomial ring
  • t, a rational number, a coefficient
• Outputs:
  • an ideal,

i1 : R = QQ[x,y];

i2 : I = monomialIdeal(y^2,x^3);

o2 : MonomialIdeal of R

i3 : multiplierIdeal(I,5/6)

o3 = monomialIdeal (x, y)

o3 : MonomialIdeal of R

i4 : J = monomialIdeal(x^8,y^6); -- Example 2 of [Howald 2000]

o4 : MonomialIdeal of R

i5 : multiplierIdeal(J,1)

                     6   5    4 2   2 3     4   5
o5 = monomialIdeal (x , x y, x y , x y , x*y , y )

o5 : MonomialIdeal of R

multiplier ideal of a hyperplane arrangement

• Usage:

  multiplierIdeal(A,m,s)

• Inputs:
  • A, a central hyperplane arrangement, a central hyperplane arrangement
  • m, a list, a list of weights for the hyperplanes in $A$ (optional)
  • s, a number, a coefficient
• Outputs:
  • an ideal,

i6 : R = QQ[x,y,z];

i7 : f = toList factor((x^2 - y^2)*(x^2 - z^2)*(y^2 - z^2)*z) / first;

i8 : A = arrangement f;

i9 : multiplierIdeal(A,3/7)

o9 = ideal (z, y, x)

o9 : Ideal of R

multiplier ideal of monomial space curve

• Usage:

  I = multiplierIdeal(R,n,t)

• Inputs:
  • R, a ring
  • n, a list, a list of three integers
  • t, a rational number
• Outputs:
  • I, an ideal

Computes the multiplier ideal of the space curve $C$ parametrized by $(t^a,t^b,t^c)$ given by $n=(a,b,c)$.

i10 : R = QQ[x,y,z];

i11 : n = {2,3,4};

i12 : t = 5/2;

i13 : I = multiplierIdeal(R,n,t)

              2         2
o13 = ideal (y  - x*z, x  - z)

o13 : Ideal of R

multiplier ideal of a generic determinantal ideal

• Usage:

  multiplierIdeal(R,L,r,t)

• Inputs:
  • R, a ring, a ring
  • L, a list, dimensions $\{m,n\}$ of a matrix
  • r, an integer, the size of minors generating the determinantal ideal
  • t, a rational number, a coefficient
• Outputs:
  • an ideal,

i14 : x = symbol x;

i15 : R = QQ[x_1..x_20];

i16 : X = genericMatrix(R,4,5);

              4      5
o16 : Matrix R  <-- R

i17 : multiplierIdeal(X,2,5/7)

o17 = ideal 1

o17 : Ideal of R