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, a ring element,
- L, a list, nonempty, of RingElements
Outputs:
- m, a module, the module of logarithmic vector fields for I or ideal(f)
- n, a module, 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

Ways to use derlog :

derlog(Ideal)
derlog(RingElement)

For the programmer

The object derlog is a method function.

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

Synopsis

Description

See also

Ways to use derlog :

For the programmer