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) 2 2 2 2 o14 = {monomialIdeal(x x ), monomialIdeal (x x , x ), monomialIdeal (x x , x )} 1 2 2 3 3 1 2 3 o14 : List |
i15 : regStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. The zero ideals were extracted from the sample before reporting the regularity statistics. o15 = (3, .816497) o15 : Sequence |
i16 : CMStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. 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, 0 are the zero ideal. o17 = (3.33333, 2.0548) o17 : Sequence |
i18 : dimStats(ideals, Verbose=>true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. o18 = (1.66667, .471405) o18 : Sequence |
i19 : borelFixedStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. 0 out of 3 monomial ideals in the given sample are Borel-fixed. o19 = 0 o19 : QQ |
i20 : mingenStats(ideals, Verbose=>true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. o20 = (1.66667, .471405, 2.66667, .471405) o20 : Sequence |
i21 : bettiStats(ideals, Verbose => true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. 0 1 2 0 1 2 1 2 o21 = (total: 1 1.33333 .666667, total: 1 1.66667 .666667, total: 1.2879 .942809) 0: 1 . . 0: 1 . . 1: .816497 .471405 1: . .666667 .333333 1: . 1 .333333 2: .471405 . 2: . .666667 . 2: . .666667 . 3: . .471405 3: . . .333333 3: . . .333333 o21 : Sequence |
i22 : M = randomMonomialSets(n,D,p,N); |
i23 : idealsFromGeneratingSets(M, Verbose => true) There are 3 ideals in this sample. Of those, 0 are the zero ideal. 2 2 2 2 2 o23 = {monomialIdeal (x x , x x , x ), monomialIdeal (x , x ), monomialIdeal x } 1 2 2 3 3 1 3 1 o23 : List |