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multIdeal -- compute a multiplier ideal

Synopsis

Description

The multiplier ideals of an given ideal depend on a nonnegative real parameter. This method computes the multiplier ideals of the defining ideal of a hyperplane arrangement, optionally with multiplicities m. This uses the explicit formula of M. Mustata [TAMS 358 (2006), no 11, 5015--5023], as simplified by Z. Teitler [PAMS 136 (2008), no 5, 1902--1913].

One can compute directly:

i1 : A = typeA(3);
i2 : hilbertSeries multIdeal(3,A)

           18
      1 - T
o2 = --------
            4
     (1 - T)

o2 : Expression of class Divide
Since the multiplier ideal is a locally constant function of its real parameter, one test to see at what values it changes:
i3 : H = new MutableHashTable

o3 = MutableHashTable{}

o3 : MutableHashTable
i4 : scan(40,i -> (
                         s := i/20.;
                         I := multIdeal(s,A);
                         if not H#?I then H#I = {s} else H#I = H#I|{s}));
i5 : netList sort values H -- values of s giving same multiplier ideal

     +---+----+---+----+---+----+---+----+---+----+---+----+---+----+
o5 = |0  |.05 |.1 |.15 |.2 |.25 |.3 |    |   |    |   |    |   |    |
     +---+----+---+----+---+----+---+----+---+----+---+----+---+----+
     |.35|.4  |.45|.5  |.55|.6  |.65|    |   |    |   |    |   |    |
     +---+----+---+----+---+----+---+----+---+----+---+----+---+----+
     |.7 |.75 |.8 |.85 |.9 |.95 |   |    |   |    |   |    |   |    |
     +---+----+---+----+---+----+---+----+---+----+---+----+---+----+
     |1  |1.05|1.1|1.15|1.2|1.25|1.3|1.35|1.4|1.45|1.5|1.55|1.6|1.65|
     +---+----+---+----+---+----+---+----+---+----+---+----+---+----+
     |1.7|1.75|1.8|1.85|1.9|1.95|   |    |   |    |   |    |   |    |
     +---+----+---+----+---+----+---+----+---+----+---+----+---+----+

Ways to use multIdeal :