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);
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i2 : hilbertSeries multIdeal(3,A)
18
1 - T
o2 = --------
4
(1 - T)
o2 : Expression of class Divide
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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
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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}));
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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| | | | | | | | |
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+
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