homogeneous cordinates

Hello again friends
I have a problem in algebraic geometry
I have a vector field in Cartesian coordinates.

I know also that the ring C ³ in homogeneous coordinates is P ²
ie (X, Y, Z) can actually be (x / z, y / z, 1)

I want to get the vector field in these new coordinates then

is there any lib who can help me?


proc transf (matrix a)
(
int n = nrows (a);
dual matrix [n] [n];

dual [1,1] = a [2,1] / / transform the field in the dual form
dual [2,1] = -1 * a [1,1];

return (dual);
)

transf (dual (a));


now need to do this function so

Not clear what this should mean.
Give an explicit mathematical example.

i have one field vector w= -q(x,y)dx +p(x,y)dy

that vector is in P²(polinomial ring with x,y)

then i transform P² in C³\{0} by one equivalence relation


then i have a space (x,y,z), but i need transform the ring such that the result coordinates are in the form
(x/z:y/z:1), this is in homogeneous coordinates, because i consider the class of points that pass through
(x,y,z)
z=1 is the plane that transform C³ in C²,

i need one transformation of this type:

w'= z^(m+2) * w=q(x,y)*z^m +p(x,y)*z^m

If you define at the beginning already a ring in three variables
(x,y,z), where you have two polynomial p and q in the variables x,y,
then you can just set




Otherwise, you have to define a new ring with an additional variable,
fetch/imap your input to this ring and proceed then.

See: extendring from ring.lib and
also the command homog

If not enough, give an explicit example.
Input: p = ... q =
Output:

And where comes m = ... from ?

m is the max degree of monomials p, q

i need, in true,create a function for desomoneization and homegeneization of polynomials.

It would be really helpful if you could give an concrete example.
State what you have, how the basering looks like, and
what you want to get.

Is m the degree of p and q, or should this also be
an additional argument to a possible proc?

In general, dehomogenization i.e. passing from homogenous
coordinates X,Y,Z to affine coordinates, where Z = 1, is
obtained by substituting Z=1.

This is done by subst(q,Z,1).

The homogenization is done by homog(q,z).
If q has degree d, then homog(q,z) is
homogenous of degree d.

// ---
Or could you point to an article/textbook where this situation is considered?

C. Gorzel

thx, but i cant do the change of variables x=X/Z and y=Y/Z, that is,

i need tha t. One vector v= x²+y²
use subst(v,x,x/z)
and subst(v,y,y/z)

and the return is v= (x/z)² +(y/z)²