dimStats -- statistics on the Krull dimension of a list of objects

Synopsis

Usage:

dimStats(List)
Inputs:
- ideals, a list, of objects of type MonomialIdeal or any type to which dim can be applied.
Optional inputs:
- ShowTally => ..., default value false, optional input to choose if the tally is to be returned
- Verbose => ..., default value false, optional input to request verbose feedback
Outputs:
- a sequence, whose first entry is the average Krull dimension of a list of monomial ideals, the second entry is the standard deviation of the Krull dimension, and third entry (if option turned on) is the Krull dimension tally

Description

The function dimStats computes the average and standard deviation of the Krull dimension for a list of monomial ideals.

i1 : R=ZZ/101[a,b,c];

i2 : ideals = {monomialIdeal"a3,b,c2", monomialIdeal"a3,b,ac"}

                      3      2                   3
o2 = {monomialIdeal (a , b, c ), monomialIdeal (a , b, a*c)}

o2 : List

i3 : dimStats(ideals)

o3 = (.5, .5)

o3 : Sequence

The following examples use the existing functions randomMonomialSets and idealsFromGeneratingSets or randomMonomialIdeals to automatically generate a list of ideals, rather than creating the list manually:

i4 : ideals = idealsFromGeneratingSets(randomMonomialSets(4,3,1.0,3))

o4 = {monomialIdeal (x , x , x , x ), monomialIdeal (x , x , x , x ),
                      1   2   3   4                   1   2   3   4  
     ------------------------------------------------------------------------
     monomialIdeal (x , x , x , x )}
                     1   2   3   4

o4 : List

i5 : dimStats(ideals)

o5 = (0, 0)

o5 : Sequence

i6 : ideals = randomMonomialIdeals(4,3,1.0,3)

o6 = {monomialIdeal (x , x , x , x ), monomialIdeal (x , x , x , x ),
                      1   2   3   4                   1   2   3   4  
     ------------------------------------------------------------------------
     monomialIdeal (x , x , x , x )}
                     1   2   3   4

o6 : List

i7 : dimStats(ideals)

o7 = (0, 0)

o7 : Sequence

i8 : ideals = idealsFromGeneratingSets(randomMonomialSets(3,7,0.01,10))

                     4 2                     5   2 4                    
o8 = {monomialIdeal(x x x ), monomialIdeal (x , x x ), monomialIdeal (),
                     1 2 3                   1   1 2                    
     ------------------------------------------------------------------------
                                       5         4                  2 3  
     monomialIdeal (), monomialIdeal (x x x , x x ), monomialIdeal(x x ),
                                       1 2 3   1 3                  2 3  
     ------------------------------------------------------------------------
                    2                      2 3                    
     monomialIdeal(x x ), monomialIdeal(x x x ), monomialIdeal (),
                    1 3                  1 2 3                    
     ------------------------------------------------------------------------
                    4 3
     monomialIdeal(x x )}
                    2 3

o8 : List

i9 : dimStats(ideals)

o9 = (2.3, .458258)

o9 : Sequence

i10 : ideals = randomMonomialIdeals(5,7,0.05,8)

                       2     2 3   5     2           3           2     3 
o10 = {monomialIdeal (x x , x x , x x , x x x , x x x x , x x x x , x x ,
                       2 3   1 3   1 4   1 3 4   1 2 3 4   1 2 3 4   3 4 
      -----------------------------------------------------------------------
       2 5   4       4                   4       4       4 2     3 2     4 2 
      x x , x x x , x x x , x x x x , x x x , x x x x , x x x , x x x , x x ,
       1 4   1 2 5   1 3 5   1 2 3 5   1 3 5   2 3 4 5   1 4 5   3 4 5   2 5 
      -----------------------------------------------------------------------
       2 2 2     2 2   2 2   2   2 2 2     4 2     3   2   3   3   3       4 
      x x x , x x x , x x x x , x x x , x x x , x x , x x x , x x x , x x x ,
       1 3 5   2 3 5   1 2 4 5   1 4 5   2 4 5   2 5   1 3 5   3 4 5   3 4 5 
      -----------------------------------------------------------------------
       2 4                   3     2 3   7       2   3   4     6             
      x x ), monomialIdeal (x x , x x , x , x x x , x , x x , x x , x x x x ,
       4 5                   1 2   1 2   2   1 2 3   3   1 4   2 4   1 2 3 4 
      -----------------------------------------------------------------------
       2   2   2 2 2     3           2 2   2   2   4   2 4       5   2 5 
      x x x , x x x , x x , x x x , x x x x , x x x , x x , x x x , x x ,
       1 2 4   2 3 4   1 4   2 4 5   1 3 4 5   1 3 5   4 5   1 2 5   2 5 
      -----------------------------------------------------------------------
       2 5                   4   6   3         2   2 4     2 2 2   2 3 
      x x ), monomialIdeal (x , x , x x , x x x , x x x , x x x , x x ,
       3 5                   1   2   1 3   1 2 3   1 3 4   1 3 4   2 4 
      -----------------------------------------------------------------------
       2     3   3 2     2       4         4       3   2     3 3       3   
      x x x x , x x x , x x x , x x x , x x x x , x x x x , x x x , x x x ,
       1 2 3 4   1 2 5   2 3 5   2 4 5   1 3 4 5   1 2 4 5   1 4 5   3 4 5 
      -----------------------------------------------------------------------
         4     4 2   3   2         2   3 2 2       3     3     2 3   2 4 
      x x x , x x , x x x , x x x x , x x x , x x x , x x , x x x , x x ,
       2 4 5   3 5   1 4 5   2 3 4 5   3 4 5   1 2 5   3 5   1 4 5   1 5 
      -----------------------------------------------------------------------
         6                   6   3 3   2     2 2       5     5 2   2   4 
      x x ), monomialIdeal (x , x x , x x , x x x , x x x , x x , x x x ,
       2 5                   1   2 3   2 4   1 3 4   1 3 4   3 4   1 2 4 
      -----------------------------------------------------------------------
         5     2     4     2     2   2           2       5     2  
      x x , x x x , x x , x x , x x x x , x x x x x , x x x , x ),
       3 4   1 2 5   2 5   3 5   1 3 4 5   1 2 3 4 5   2 4 5   5  
      -----------------------------------------------------------------------
                          3 4   4 2     3 3     3   2   4 2   2 3     2 3 
      monomialIdeal (x , x x , x x x , x x x , x x x , x x , x x , x x x ,
                      2   1 3   1 3 4   1 3 4   1 3 4   3 4   1 4   1 3 4 
      -----------------------------------------------------------------------
             4     2                   4       5             2   3     6   
      x x , x x , x ), monomialIdeal (x x , x x , x x x , x x , x x , x x ,
       3 5   4 5   5                   1 2   1 2   1 2 3   1 3   2 4   3 4 
      -----------------------------------------------------------------------
       2 2       3     4     6   3     5       2       2 2 2       2   2 2 
      x x , x x x , x x , x x , x x , x x x , x x x , x x x , x x x , x x ,
       1 4   1 2 4   1 4   2 4   1 5   2 3 5   3 4 5   1 2 5   1 4 5   4 5 
      -----------------------------------------------------------------------
       2 4                         4 2   2 3     2 2     2 2   3 2   5 
      x x ), monomialIdeal (x x , x x , x x , x x x , x x x , x x , x ,
       2 5                   1 3   2 3   2 3   1 2 4   2 3 4   3 4   4 
      -----------------------------------------------------------------------
       3         2     2 4     4   2   2 2 2     3                     
      x x x , x x x , x x x , x x x , x x x , x x ), monomialIdeal (x ,
       1 2 5   1 2 5   3 4 5   3 4 5   1 4 5   4 5                   2 
      -----------------------------------------------------------------------
       2         2       3   7     4 2   4 3     4
      x x x , x x x , x x , x , x x x , x x , x x )}
       1 3 4   1 3 4   3 4   4   1 3 5   4 5   4 5

o10 : List

i11 : dimStats(ideals)

o11 = (1.75, .433013)

o11 : Sequence

i12 : ideals = idealsFromGeneratingSets(randomMonomialSets(5,7,1,10))

                        3 2                      2 4                    5    
o12 = {monomialIdeal(x x x x ), monomialIdeal(x x x ), monomialIdeal(x x x ),
                      1 2 3 4                  3 4 5                  2 3 5  
      -----------------------------------------------------------------------
                       2 3                  2       2  
      monomialIdeal(x x x ), monomialIdeal(x x x x x ),
                     1 3 4                  1 2 3 4 5  
      -----------------------------------------------------------------------
                     2 2 3                  2 4                  3 4  
      monomialIdeal(x x x ), monomialIdeal(x x ), monomialIdeal(x x ),
                     1 3 4                  1 2                  3 4  
      -----------------------------------------------------------------------
                     6                 2   3
      monomialIdeal x , monomialIdeal(x x x )}
                     1                 1 2 5

o12 : List

i13 : dimStats(ideals)

o13 = (4, 0)

o13 : Sequence

Ways to use dimStats :

dimStats(List)

For the programmer

The object dimStats is a method function with options.