# derlog -- compute the logarithmic (tangent) vector fields to an ideal

## Synopsis

• Usage:
m=derlog(I)
m=derlog(f)
n=derlogH(L)
n=derlogH(f)
• Inputs:
• I, an ideal,
• f, ,
• L, a list, nonempty, of RingElements
• Outputs:
• m, , the module of logarithmic vector fields for I or ideal(f)
• n, , the module of vector fields that annihilates f or the entries of L

## Description

derlog computes the module of logarithmic vector fields to an affine variety defined by I or f; these are the ambient vector fields tangent to the variety.

derlogH computes the module of ambient vector fields tangent to all level sets of f or of the entries of L.

Note that derlog(I)=der(I,I) and derlogH(L)=der(L,0); see der.

 i1 : R=QQ[x,y,z]; i2 : f=x*y-z^2; i3 : derlog(ideal (f)) o3 = image | 2x 0 2z 0 | | 0 2y 0 2z | | z z y x | 3 o3 : R-module, submodule of R i4 : derlogH(f) o4 = image | x 2z 0 | | -y 0 2z | | 0 y x | 3 o4 : R-module, submodule of R i5 : dH=derlogH({f}) o5 = image | x 2z 0 | | -y 0 2z | | 0 y x | 3 o5 : R-module, submodule of R

Although every element of dH annihilates f, they do not annihilate the ideal generated by f:

 i6 : applyVectorField(dH,f) o6 = ideal (0, 0, 0) o6 : Ideal of R i7 : applyVectorField(dH,ideal(f)) 3 2 2 2 2 o7 = ideal (x*y*z - z , x*y - y*z , x y - x*z ) o7 : Ideal of R