Next: 3.2 The global homogeneous Up: 3. Finding a regular Previous: 3. Finding a regular

## 3.1 The local case

At this point the local case is much easier to handle. For the invertibility of and it is sufficient that its constant parts have full rank. Hence, we can restrict ourself to and with Xc and Yc denoting the constant parts of the corresponding matrices and . It follows that and are homogeneous polynomials of degree m and n respectively in the indeterminates . We have to find a point not lying on the projective hypersurface determined by .

Lemma 3..1   M and M' are isomorphic if and only if .

Proof: This statement is obvious.

Now, assuming that , we can recursively insert m+n+1 different integer values for any ci on which F depends. As for one of these values F does not vanish and we can repeat the procedure with F(ci=pi) instead of F. Choosing at the end arbitrary (for example 0) values for the free ci (Those on which F does not depend!) we obtain the desired point P.

Next: 3.2 The global homogeneous Up: 3. Finding a regular Previous: 3. Finding a regular
| ZCA Home | Reports |