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RandomMonomialIdeals :: degStats(...,Verbose=>...)

degStats(...,Verbose=>...) -- optional input to request verbose feedback

Description

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

Further information

See also

Functions with optional argument named Verbose :