  Let A= (a_{ij}) be a non-negative integer k x k matrix. A is a homogeneous
matrix if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes.
We ask: If A is a homogeneous matrix and if F is a form in C[x_1, \dots x_n]
with deg(F) = trace(A), what is the least integer, s(A), so that F = det M_1 +
... + det M_{s(A)}, where the M_i's are k x k matrices of forms with degree
matrix A? We consider this problem for n>3 and we prove that s(A) is at most
k^{n-3} and s(A) <k^{n-3} in infinitely many cases. However s(A) = k^{n-3} when
the entries of A are large with respect to k.
