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

## Synopsis

• Function: logCohomology
• Usage:
logCohomology f
• Inputs:
• f, , polynomial in two variables
• Outputs:
• , 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