der(CentralArrangement,List) -- compute the module of logarithmic derivations

Synopsis

Function: der
Usage:

der(A, m)

der(A)
Inputs:
- A, a central hyperplane arrangement, a central arrangement of hyperplanes
- m, a list, an optional list of multiplicities, one for each hyperplane
Optional inputs:
- Strategy => a symbol, default value null, that specifies the algorithm. If an arrangement has (squarefree) defining polynomial $Q$, then the logarithmic derivations are those derivations $D$ for which $D(Q)$ is in the ideal $(Q)$. The Popescu strategy assumes that the arrangement is simple and implements this definition. By contrast, the default strategy treats all arrangements as multiarrangements.
Outputs:
- a matrix, whose image is the module of logarithmic derivations corresponding to the (multi)arrangement ${\mathcal A}$; see below.

Description

The module of logarithmic derivations of an arrangement defined over a ring $S$ is, by definition, the submodule of $S$-derivations $D$ 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.

In this package, we grade derivations so that a constant coefficient derivation (i.e. a derivation $D$ which takes linear forms to constants) has degree 0. In the literature, this is often called polynomial degree.

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 the ideal $(f_i^{m_i})$ for each linear form $f_i$. See Günter M. Ziegler, Multiarrangements of hyperplanes and their freeness, in Singularities (Iowa City, IA, 1986), 345-359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989.

The $j$th column of the output matrix expresses the $j$th 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 0      0            |
     {1} | -1 -x+y   0            |
     {1} | -1 -x+z   -xy+xz+yz-z2 |
     {1} | -1 y      0            |
     {1} | -1 z      yz-z2        |
     {1} | -1 -x+y+z xz-z2        |

             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 ${\mathcal A}$ is treated as if it were essential: that is, the intersection of all the hyperplanes is the origin. So, the rank of the matrix produced by der equals the rank of the arrangement. For instance, although the $A_3$ arrangement is not essential, der will produce a rank 3 matrix.

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)

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

o4 : QQ[x ..x ]-module, free, degrees {1, 3, 5, 7}
         1   4

A hyperplane arrangement ${\mathcal A}$ is free if the module of derivations is a free $S$-module. 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 : der A

o7 = {1} | -1 0  0    0      |
     {1} | -1 z  -z   -x-y-z |
     {1} | -1 -y -x-z 0      |
     {1} | -1 0  -z   -y     |

             4      4
o7 : Matrix R  <-- R

i8 : betti res prune image der A

            0 1
o8 = total: 4 1
         1: 1 .
         2: 3 1

o8 : BettiTally

The Popescu strategy produces a different presentation of the module of logarithm derivations. For instance, in the following example, the first three rows of column 0 means that $x\frac{\partial}{\partial x} + y\frac{\partial}{\partial y} + z\frac{\partial}{\partial z}$ is a logarithmic derivation of $\mathcal A$, and the last row of column 0 means that applying this derivation to $xyz(x+y+z)$ produces $4xyz(x+y+z)$.

i9 : der(A, Strategy => Popescu)

o9 = {0} | x 0   0     |
     {0} | y -yz yz    |
     {0} | z yz  xz+z2 |
     {1} | 4 y-z x+3z  |

             4      3
o9 : Matrix R  <-- R

If a list of multiplicities is not provided, the occurrences of each hyperplane are counted:

i10 : R = QQ[x,y]

o10 = R

o10 : PolynomialRing

i11 : prune image der arrangement {x,y,x-y,y-x,y,2*x}   -- rank 2 => free

       2
o11 = R

o11 : R-module, free, degrees {2:3}

i12 : prune image der(arrangement {x,y,x-y}, {2,2,2})  -- same

       2
o12 = R

o12 : R-module, free, degrees {2:3}

Ways to use this method: