der(A, m)
der(A)
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), 345359, 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.


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.


A hyperplane arrangement ${\mathcal A}$ is free if the module of derivations is a free $S$module. Not all arrangements are free.




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 $4xyz(x+y+z)$.

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


