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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.

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.

i1 : prune image der typeA(3)

                          3
o1 = (QQ [x , x , x , x ])
           1   2   3   4

o1 : QQ [x , x , x , x ]-module, free, degrees {1, 2, 3}
          1   2   3   4
i2 : prune image der typeB(4) -- A is said to be free if der(A) is a free module

                          4
o2 = (QQ [x , x , x , x ])
           1   2   3   4

o2 : QQ [x , x , x , x ]-module, free, degrees {1, 3, 5, 7}
          1   2   3   4
not all arrangements are free:
i3 : R = QQ[x,y,z];
i4 : A = arrangement {x,y,z,x+y+z}

o4 = A

o4 : Hyperplane Arrangement 
i5 : betti res prune image der A

            0 1
o5 = total: 4 1
         1: 1 .
         2: 3 1

o5 : BettiTally
If a list of multiplicities is not provided, the occurrences of each hyperplane are counted:
i6 : R = QQ[x,y]

o6 = R

o6 : PolynomialRing
i7 : prune image der arrangement {x,y,x-y,y-x,y,2*x}   -- rank 2 => free

      2
o7 = R

o7 : R-module, free, degrees {3, 3}
i8 : prune image der(arrangement {x,y,x-y}, {2,2,2})  -- same thing

      2
o8 = R

o8 : R-module, free, degrees {3, 3}

Ways to use der :