# pdimStats -- statistics on projective dimension of a list of objects

## Synopsis

• Usage:
pdimStats(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 mean projective dimension, the second entry is the standard deviation of the projective dimension, and third entry (if option turned on) is the projective dimension tally for quotient rings of ideals in the list ideals.

## Description

The function pdimStats computes the mean and standard deviation of the projective dimension of elements in the list:

 i1 : R=ZZ/101[a,b,c]; i2 : ideals = {monomialIdeal(a^3,b,c^2), monomialIdeal(a^3,b,a*c)} 3 2 3 o2 = {monomialIdeal (a , b, c ), monomialIdeal (a , b, a*c)} o2 : List i3 : pdimStats(ideals) o3 = (3, 0) o3 : Sequence

The function can also output the projective dimension tally as follows:

 i4 : R=ZZ/101[a,b,c]; i5 : ideals = {monomialIdeal(a,c),monomialIdeal(b),monomialIdeal(a^2*b,b^2)} 2 2 o5 = {monomialIdeal (a, c), monomialIdeal b, monomialIdeal (a b, b )} o5 : List i6 : pdimStats(ideals, ShowTally=>true) o6 = (1.66667, .471405, Tally{1 => 1}) 2 => 2 o6 : Sequence

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

 i7 : ideals = randomMonomialIdeals(4,3,1.0,3) o7 = {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 o7 : List i8 : pdimStats(ideals) o8 = (4, 0) o8 : Sequence i9 : ideals = randomMonomialIdeals(4,6,0.01,10) 2 3 2 3 o9 = {monomialIdeal(x x x ), monomialIdeal (x x x , x x x ), monomialIdeal 2 3 4 1 2 4 1 3 4 ------------------------------------------------------------------------ 5 3 2 2 2 (), monomialIdeal (x x , x x , x x ), monomialIdeal(x x ), 2 3 2 3 1 4 2 3 ------------------------------------------------------------------------ 2 2 5 4 2 2 monomialIdeal(x x ), monomialIdeal (x x , x x x , x x x x ), 1 4 1 3 2 3 4 1 2 3 4 ------------------------------------------------------------------------ 2 3 2 2 2 monomialIdeal(x x ), monomialIdeal (x x x , x x x , x ), 2 3 1 3 4 1 3 4 4 ------------------------------------------------------------------------ 3 monomialIdeal(x x x )} 2 3 4 o9 : List i10 : pdimStats(ideals) o10 = (1.6, 1.0198) o10 : Sequence