3.1 The local case

At this point the local case is much easier to handle. For the invertibility of $X(\underline{f})$ and $Y(\underline{f})$ it is sufficient that its constant parts have full rank. Hence, we can restrict ourself to $X^c(\underline{c})=\sum_{i=1}^r c_iX^c_i$ and $Y^c(\underline{c})=\sum_{i=1}^r c_iY^c_i$ with X^c and Y^c denoting the constant parts of the corresponding matrices and $c_i\in k$. It follows that $det(X^c(\underline{c}))$ and $det(Y^c(\underline{c}))$ are homogeneous polynomials of degree m and n respectively in the indeterminates $c_i,\ i=1,\ldots,r$. We have to find a point $P_{\underline{c}}$ not lying on the projective hypersurface determined by $F(\underline{c})=det(X^c(\underline{c}))det(Y^c(\underline{c}))$.

Lemma 3..1   M and M' are isomorphic if and only if $F(\underline{c})\not=0$.

Proof: This statement is obvious.

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