Some of the functions that use this option by default exclude zero ideals when computing statistics on a set of ideals, while others do not. If Verbose => true, then the functions will display this type of additional informational. The default value is false.
i1 : n=3;D=3;p=0.0;N=3; |
i5 : ideals = randomMonomialIdeals(n,D,p,N)
o5 = {monomialIdeal (), monomialIdeal (), monomialIdeal ()}
o5 : List
|
i6 : regStats(ideals) o6 = (-infinity, 0) o6 : Sequence |
i7 : CMStats(ideals) o7 = 1 o7 : QQ |
In the examples above, one may wonder, for example, why 3 out of 3 ideals in the list are Cohen-Macaulay. In order to view the additional information, set Verbose => true:
i8 : regStats(ideals, Verbose => true) All ideals in this list are the zero ideal. o8 = (-infinity, 0) o8 : Sequence |
i9 : CMStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 3 are the zero ideal. The zero ideals are included in the reported count of Cohen-Macaulay quotient rings. 3 out of 3 ideals in the given sample are Cohen-Macaulay. o9 = 1 o9 : QQ |
Other functions that have this option are as follows. Let us look at a list of nontrivial ideals to see more interesting statistics.
i10 : n=3;D=3;p=0.1;N=3; |
i14 : ideals = randomMonomialIdeals(n,D,p,N)
3
o14 = {monomialIdeal (x , x , x ), monomialIdeal (), monomialIdeal (x x ,
1 2 3 2 3
-----------------------------------------------------------------------
2
x )}
3
o14 : List
|
i15 : regStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 1 are the zero ideal. The zero ideals were extracted from the sample before reporting the regularity statistics. o15 = (2.5, .5) o15 : Sequence |
i16 : CMStats(ideals, Verbose => true)
There are 3 ideals in this sample. Of those, 1 are the zero ideal.
The zero ideals are included in the reported count of Cohen-Macaulay quotient rings.
2 out of 3 ideals in the given sample are Cohen-Macaulay.
2
o16 = -
3
o16 : QQ
|
i17 : degStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 1 are the zero ideal. The degree statistics do include those for the zero ideals. o17 = (1.66667, .942809) o17 : Sequence |
i18 : dimStats(ideals, Verbose=>true) There are 3 ideals in this sample. Of those, 1 are the zero ideal. The Krull dimension statistics do include those for the zero ideals. o18 = (1.66667, 1.24722) o18 : Sequence |
i19 : borelFixedStats(ideals, Verbose => true)
There are 3 ideals in this sample. Of those, 1 are the zero ideal.
The zero ideals are included in the reported count of Borel-fixed monomial ideals.
1 out of 3 monomial ideals in the given sample are Borel-fixed.
1
o19 = -
3
o19 : QQ
|
i20 : mingenStats(ideals, Verbose=>true) There are 3 ideals in this sample. Of those, 1 are the zero ideal. The statistics returned (mean and standard deviation of # of min gens and mean and standard deviation of degree complexity) do NOT include those for the zero ideals. o20 = (2.5, .5, 2.5, .5) o20 : Sequence |
i21 : bettiStats(ideals, Verbose => true)
There are 3 ideals in this sample. Of those, 1 are the zero ideal.
The Betti statistics do include those for the zero ideals.
0 1 2 3 0 1 2 3
o21 = (total: 1 1 1 .333333, total: 1 1.66667 1.33333 .333333,
0: 1 .333333 .333333 . 0: 1 .666667 .333333 .
1: . .333333 .333333 . 1: . .666667 .333333 .
2: . .333333 .333333 .333333 2: . .333333 .666667 .333333
-----------------------------------------------------------------------
1 2 3
total: 2.35702 1.88562 .471405)
0: .942809 .471405 .
1: .942809 .471405 .
2: .471405 .942809 .471405
o21 : Sequence
|
i22 : M = randomMonomialSets(n,D,p,N); |
i23 : idealsFromGeneratingSets(M, Verbose => true)
There are 3 ideals in this sample. Of those, 1 are the zero ideal.
2
o23 = {monomialIdeal (x x , x ), monomialIdeal (), monomialIdeal(x x )}
1 2 3 2 3
o23 : List
|