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

## Synopsis

• Usage:
dimStats(List)
• Inputs:
• 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:
• , 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 .