logCohomology(RingElement) -- logarithmic cohomology groups in two variables

Synopsis

Function: logCohomology
Usage:

logCohomology f
Inputs:
- f, polynomial in two variables
Outputs:
- a hash table, with entries {VResolution, Input, TransferCycles, CohomologyGroups, PreCycles, OmegaRes, LocalizeMap, BFunction}

Description

For a polynomial f in two variables executes the algorithm described in Castro-Jimenez and Takayama "The Computation of the Logarithmic Cohomology for Plane Curves" (arXiv:0712.0001).

i1 : S=QQ[x,y];

i2 : f=x*y*(x-y);

i3 : logCohomology(f)

o3 = HashTable{BFunction => (s - 1)                                                                    }
                                                    1
               CohomologyGroups => HashTable{0 => QQ }
                                                    3
                                             1 => QQ
                                                    2
                                             2 => QQ
               Input => cokernel | xdx+ydy+3 xydy-y2dy+x-2y |
               LocalizeMap => LocMap
                                             1                       2                       1
               OmegaRes => (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- 0
                                                                                                    
                           0                       1                       2                       3
               PreCycles => HashTable{0 => | x |       }
                                           | 1 |
                                      1 => | -x -y 0  |
                                           | 0  0  -1 |
                                           | 0  0  0  |
                                      2 => | -y -x |
               TransferCycles => HashTable{0 => | -x2y+xy2 |    }
                                           1 => | xy y2 -xy+y2 |
                                                | x2 xy 0      |
                                           2 => | -y -x |
                                                1                       3                       2
               VResolution => (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- 0
                                                                                                       
                              0                       1                       2                       3

o3 : HashTable

Ways to use this method: