For example, count the monomial ideals in $\mathbb{Q}[x,y,z]$, generated in degrees up to 5, whose Hilbert function begins with $\{1, 3, 6, 5, 4, 4\}$:
i1 : R = QQ[x,y,z]; L = {1, 3, 6, 5, 4, 4};
|
i3 : M = monomialIdealsWithHilbertFunction(L, R); #M o4 = 306 |
i5 : netList take(M, 5)
+-------------------------------------------------+
| 2 2 2 2 2 |
o5 = |monomialIdeal (x y, x*y , x z, x*z , y*z ) |
+-------------------------------------------------+
| 2 2 2 2 2 |
|monomialIdeal (x y, x*y , y z, x*z , y*z ) |
+-------------------------------------------------+
| 2 2 3 2 2 |
|monomialIdeal (x y, x*y , x*y*z, y z, x*z , y*z )|
+-------------------------------------------------+
| 2 2 2 2 4 |
|monomialIdeal (x y, x*y , x*y*z, x*z , y*z , z ) |
+-------------------------------------------------+
| 2 2 2 2 3 |
|monomialIdeal (x y, x*y , x*z , y*z , z ) |
+-------------------------------------------------+
|
By default, the degrees of generators are bounded by the length of $L$. A lower bound can be set manually with the BoundGenerators option.
i6 : M = monomialIdealsWithHilbertFunction(L, R, BoundGenerators => 3); #M o7 = 57 |
i8 : netList take(M, 5)
+----------------------------------------+
| 3 2 3 2 |
o8 = |monomialIdeal (x , x*y , y , x*y*z, y z)|
+----------------------------------------+
| 3 2 3 2 |
|monomialIdeal (x , x y, y , x*y*z, y z) |
+----------------------------------------+
| 3 2 3 2 |
|monomialIdeal (x , x*y , y , x z, x*y*z)|
+----------------------------------------+
| 3 2 3 2 |
|monomialIdeal (x , x y, y , x z, x*y*z) |
+----------------------------------------+
| 3 2 3 2 2 |
|monomialIdeal (x , x*y , y , x z, y z) |
+----------------------------------------+
|
There is also an option to enumerate squarefree monomial ideals only.
i9 : S = QQ[a..f] o9 = S o9 : PolynomialRing |
i10 : I = monomialIdealsWithHilbertFunction({1, 6, 19, 45, 84}, S, SquareFree => true); #I
o11 = 60
|
i12 : first random I
o12 = monomialIdeal (a*b, b*d, a*c*d*e, a*c*d*f, a*c*e*f, b*c*e*f, a*d*e*f,
-----------------------------------------------------------------------
c*d*e*f)
o12 : MonomialIdeal of S
|
To specify the total number of minimal generators, use FirstBetti.
i13 : #monomialIdealsWithHilbertFunction({1, 3, 6, 5, 4, 4}, R, FirstBetti => 5)
o13 = 57
|
i14 : #monomialIdealsWithHilbertFunction({1, 3, 6, 5, 4, 4}, R, FirstBetti => 6)
o14 = 174
|
Alternatively, specify the number of minimal generators in each degree using GradedBettis. The length of the list of graded (first) Betti numbers should match the length of the partial Hilbert function.
i15 : #monomialIdealsWithHilbertFunction({1, 3, 4, 2, 1}, R, GradedBettis => {0, 0, 2, 2, 1})
o15 = 30
|
Notice that the GradedBettis option totally constrains the degrees of generators already, so do not use it with the BoundGenerators option.
You can combine BoundGenerators with FirstBetti, however, since FirstBetti does not constrain degrees.
i16 : #monomialIdealsWithHilbertFunction({1, 3, 6, 7, 6, 5, 4, 4, 4}, R, FirstBetti => 6, BoundGenerators => 5)
o16 = 654
|
i17 : #monomialIdealsWithHilbertFunction({1, 3, 6, 7, 6, 5, 4, 4, 4}, R, FirstBetti => 6, BoundGenerators => 4)
o17 = 60
|
The SquareFree option can be used with any of the other options.
i18 : #monomialIdealsWithHilbertFunction({1, 4, 7, 10, 13}, S, SquareFree => true, FirstBetti => 5)
o18 = 240
|
i19 : #monomialIdealsWithHilbertFunction({1, 4, 7, 10, 13}, S, SquareFree => true, BoundGenerators => 3)
o19 = 300
|
i20 : #monomialIdealsWithHilbertFunction({1, 4, 7, 10, 13}, S, SquareFree => true, GradedBettis => {0, 2, 3, 1, 0})
o20 = 60
|
The object monomialIdealsWithHilbertFunction is a method function with options.