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DGAlgebras :: deviationsToPoincare

deviationsToPoincare -- Computes the power series corresponding to a set of deviations.

Synopsis

Description

This command takes a HashTable of the same form output from deviations and produces the Poincare series corresponding to it. The (key,value) pairs must be of the form homologicalDegree=>number or (homologicalDegree,internalDegree)=>number. Because

i1 : R = ZZ/101[a,b,c]/ideal{a^3,b^3,c^3}

o1 = R

o1 : QuotientRing
i2 : RDevs = deviations(R,DegreeLimit=>6)

o2 = HashTable{(1, {1}) => 3}
               (2, {3}) => 3

o2 : HashTable
i3 : devPSeries = deviationsToPoincare(RDevs,DegreeLimit=>6)

        6 9      6 8      5 7     5 6     4 6     4 5     3 4    3 3     2 3
o3 = 10S T  + 18S T  + 18S T  + 3S T  + 6S T  + 9S T  + 9S T  + S T  + 3S T 
          0        0        0       0       0       0       0      0       0
     ------------------------------------------------------------------------
         2 2
     + 3S T  + 3S*T  + 1
           0       0

     ZZ[S, T ]
            0
o3 : ---------
          7
         S
i4 : pSeries = poincareN (res(coker vars R, LengthLimit=>6))

                   2 2     2 3    3 3     3 4     4 5     4 6     5 6  
o4 = 1 + 3S*T  + 3S T  + 3S T  + S T  + 9S T  + 9S T  + 6S T  + 3S T  +
             0       0       0      0       0       0       0       0  
     ------------------------------------------------------------------------
        5 7      6 8      6 9
     18S T  + 18S T  + 10S T
          0        0        0

o4 : ZZ[S, T ]
            0
i5 : substitute(devPSeries,ring pSeries) == pSeries

o5 = true

Ways to use deviationsToPoincare :

For the programmer

The object deviationsToPoincare is a method function with options.