This function gives a convenient expression of the Hilbert sequence, particularly in terms of the multiplicity sequence. For instance, if I is an ideal, then the multiplicity sequence of I appears as the top row of the table for the Hilbert sequence of gr_mGr_I.
i1 : R = QQ[x_1..x_9] o1 = R o1 : PolynomialRing |
i2 : I = minors(2, genericMatrix(R, 3, 3))
o2 = ideal (- x x + x x , - x x + x x , - x x + x x , - x x + x x , -
2 4 1 5 3 4 1 6 3 5 2 6 2 7 1 8
------------------------------------------------------------------------
x x + x x , - x x + x x , - x x + x x , - x x + x x , - x x + x x )
3 7 1 9 3 8 2 9 5 7 4 8 6 7 4 9 6 8 5 9
o2 : Ideal of R
|
i3 : multiplicitySequence I
o3 = HashTable{4 => 6 }
5 => 12
6 => 12
7 => 6
8 => 3
9 => 2
o3 : HashTable
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i4 : printHilbertSequence hilbertSequence grGr I
o4 = 0 1 2 3 4 5 6 7 8 9
+-----------------------------
9 | . . . . 6 12 12 6 3 2
8 | . . . . -18 -30 -21 -9 -4 .
7 | . . . . 19 23 10 3 . .
6 | . . . . -8 -5 -1 . . .
5 | . . . . 1 . . . . .
4 | . . . . . . . . . .
3 | . . . . . . . . . .
2 | . . . . . . . . . .
1 | . . . . . . . . . .
0 | . . . . . . . . . .
|
The object printHilbertSequence is a method function.