# isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test

## Synopsis

• Function: isInMultiplierIdeal
• Usage:
b = isInMultiplierIdeal(g,I,c)
• Inputs:
• g, , a polynomial
• I, an ideal, an ideal in a polynomial ring
• c, , coefficient (or a list of coefficients)
• Optional inputs:
• Strategy => ..., default value generalizedBFunction
• Outputs:
• b, , answers: is in the multiplier ideal J_I(c)?

## Description

Test if the given polynomial is in the multiplier ideal for given ideal and coefficient. In general, the test is cheaper than computing the whole multiplier ideal.

There are two options for strategy:
• generalizedBFunction -- via computation of the generalized Bernstein-Sato polynomial
• mGeneralizedBFunction -- via computation of the m-generalized Bernstein-Sato polynomial
See Berkesch and Leykin Algorithms for Bernstein-Sato polynomials and multiplier ideals'' for details.
 i1 : R = QQ[x_1..x_4]; i2 : isInMultiplierIdeal(x_1, ideal {x_1^3 - x_2^2, x_2^3 - x_3^2}, 31/18) o2 = false i3 : isInMultiplierIdeal(x_1*x_2, ideal {x_1^3 - x_2^2, x_2^3 - x_3^2}, 31/18) o3 = true