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HyperplaneArrangements :: der

der -- Module of logarithmic derivations

Synopsis

Description

The module of logarithmic derivations of an arrangement defined over a ring S is, by definition, the submodule of S-derivations with the property that D(f_i) is contained in the ideal generated by f_i for each linear form f_i in the arrangement.

More generally, if the linear form f_i is given a positive integer multiplicity m_i, then the logarithmic derivations are those D with the property that D(f_i) is in ideal(f_i^(m_i)) for each linear form f_i.

The jth column of the output matrix expresses the jth generator of the derivation module in terms of its value on each linear form, in order.

i1 : R = QQ[x,y,z];
i2 : der arrangement {x,y,z,x-y,x-z,y-z}

o2 = {1} | 1 -x+y+z -xy+y2       |
     {1} | 1 y      y2-yz        |
     {1} | 1 -x+y   -xy+y2+xz-yz |
     {1} | 1 -x+z   0            |
     {1} | 1 z      0            |
     {1} | 1 0      0            |

             6       3
o2 : Matrix R  <--- R

This method is implemented in such a way that any derivations of degree 0 are ignored. Equivalently, the arrangement A is forced to be essential: that is, the intersection of all the hyperplanes is the origin.

i3 : prune image der typeA(3)

                 3
o3 = (QQ[x ..x ])
          1   4

o3 : QQ[x ..x ]-module, free, degrees {1..3}
         1   4
i4 : prune image der typeB(4) -- A is said to be free if der(A) is a free module

                 4
o4 = (QQ[x ..x ])
          1   4

o4 : QQ[x ..x ]-module, free, degrees {1, 3, 5, 7}
         1   4
not all arrangements are free:
i5 : R = QQ[x,y,z];
i6 : A = arrangement {x,y,z,x+y+z}

o6 = {x, y, z, x + y + z}

o6 : Hyperplane Arrangement 
i7 : betti res prune image der A

            0 1
o7 = total: 4 1
         1: 1 .
         2: 3 1

o7 : BettiTally
If a list of multiplicities is not provided, the occurrences of each hyperplane are counted:
i8 : R = QQ[x,y]

o8 = R

o8 : PolynomialRing
i9 : prune image der arrangement {x,y,x-y,y-x,y,2*x}   -- rank 2 => free

      2
o9 = R

o9 : R-module, free, degrees {2:3}
i10 : prune image der(arrangement {x,y,x-y}, {2,2,2})  -- same thing

       2
o10 = R

o10 : R-module, free, degrees {2:3}

Ways to use der :

For the programmer

The object der is a method function with options.