# multiplierIdeal -- multiplier ideal

## multiplier ideal of a monomial ideal

• Usage:
multiplierIdeal(I,t)
• Inputs:
• I, , a monomial ideal in a polynomial ring
• t, , a coefficient
• Outputs:
Computes the multiplier ideal of $I$ with coefficient $t$ using Howald's Theorem and the package Normaliz.
 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
• m, a list, a list of weights for the hyperplanes in $A$ (optional)
• s, , a coefficient
• Outputs:
Computes the multiplier ideal of the ideal of $A$ with coefficient $s$ using the package HyperplaneArrangements.
 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,
• Outputs:

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 coefficient
• Outputs:
Computes the multiplier ideal of the ideal of $r \times r$ minors in a $m \times n$ matrix whose entries are independent variables in the ring $R$ (a generic matrix).
 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

• MonomialIdeal -- the class of all monomial ideals handled by the engine

## Ways to use multiplierIdeal :

• "multiplierIdeal(CentralArrangement,List,Number)"
• "multiplierIdeal(CentralArrangement,Number)"
• "multiplierIdeal(Matrix,ZZ,QQ)"
• "multiplierIdeal(Matrix,ZZ,ZZ)"
• "multiplierIdeal(MonomialIdeal,QQ)"
• "multiplierIdeal(MonomialIdeal,ZZ)"
• "multiplierIdeal(Ring,List,QQ)"
• "multiplierIdeal(Ring,List,ZZ)"

## For the programmer

The object multiplierIdeal is .