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 y-z -yz+z2 | {1} | 1 y 0 | {1} | 1 x-z -xz+z2 | {1} | 1 0 0 | {1} | 1 x 0 | {1} | 1 x+y-z xy-xz-yz+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 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 |
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 |
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} |
The object der is a method function with options.