# MultiplierIdeals -- Computes the jumping numbers and their ideals

## Synopsis

• Usage:
MultiplierIdeals(F,E)
• Inputs:
• F, Matrix
• E, Intersection matrix.
• Optional inputs:
• algorithm => ..., default value AlbAlvDac, Method used to compute the jumping numbers and multiplier ideals
• BiggestJN => ..., default value 2, Upper bound of the interval where we want to compute the JN.
• JumpingDivisor => ..., default value true, Show or not the jumping divisors.
• MaxIterations => ..., default value 10000, Limits the number of iterations of the Unloading algorithm.
• SmallestJN => ..., default value 0, Lower bound of the interval where we want to compute the JN.
• Outputs:
• A table that contains at least the jumping number, their multiplicities and the ideals

## Description

Starting form the divisor encoded as a matrix of dimensions 1 x m, and the intersection matrix as presented in [AAD14], the algorithm computes the jumping numbers for this ideal with their multiplicities and associated ideals in the interval (SmallestJN,BiggestJN].
 i1 : E = matrix({{ -5, 0, 1, 0, 1}, { 0, -2, 1, 0, 0}, { 1, 1, -1, 0, 0}, { 0, 0, 0, -2, 1}, { 1, 0, 0, 1, -1}}) o1 = | -5 0 1 0 1 | | 0 -2 1 0 0 | | 1 1 -1 0 0 | | 0 0 0 -2 1 | | 1 0 0 1 -1 | 5 5 o1 : Matrix ZZ <--- ZZ i2 : F = matrix({{4,5,10,5,10}}) o2 = | 4 5 10 5 10 | 1 5 o2 : Matrix ZZ <--- ZZ i3 : MultiplierIdeals(F,E,BiggestJN => 1) 1 o3 = Jumping number: - Multiplicity: 1 Multiplier ideal: | 1 1 2 1 2 | Maximal jumping divisor: {| 1 0 1 0 1 |} 2 Minimal jumping divisor: {| 1 0 1 0 1 |} 7 Jumping number: -- Multiplicity: 2 Multiplier ideal: | 2 2 4 2 4 | Maximal jumping divisor: {| 0 0 1 0 1 |} 10 Minimal jumping divisor: {| 0 0 1 0 1 |} 9 Jumping number: -- Multiplicity: 2 Multiplier ideal: | 2 3 5 3 5 | Maximal jumping divisor: {| 0 0 1 0 1 |} 10 Minimal jumping divisor: {| 0 0 1 0 1 |} Jumping number: 1 Multiplicity: 1 Multiplier ideal: | 3 3 6 3 6 | Maximal jumping divisor: {| 1 1 1 1 1 |} Minimal jumping divisor: {| 1 0 1 0 1 |} o3 : Type of HashTable

## For the programmer

The object MultiplierIdeals is .