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RandomMonomialIdeals :: dimStats

dimStats -- statistics on the Krull dimension of a list of monomial ideals

Synopsis

Description

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

o8 : List
i9 : dimStats(ideals)

o9 = (2.1, .538516)

o9 : Sequence
i10 : ideals = randomMonomialIdeals(5,7,0.05,8)

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

o10 : List
i11 : dimStats(ideals)

o11 = (1.625, .484123)

o11 : Sequence
i12 : ideals = idealsFromGeneratingSets(randomMonomialSets(5,7,1,10))

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

o12 : List
i13 : dimStats(ideals)

o13 = (4, 0)

o13 : Sequence

Note that this function can be run with a list of any objects to which dim can be applied.

Ways to use dimStats :