Expressing a general form as a sum of determinants

Let \(A= (a_{ij})\) be a non-negative integer \(k\times k\) 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 \(\mathbb {C}[x_1, \dots x_n]\) with \(deg(F) = \mathrm{trace}(A)\), what is the least integer, \(s(A)\), so that \(F = detM_1 + \cdots + detM_{s(A)}\), where the \(M_i = (F^i_{lm})\) are \(k\times k\) matrices of forms and \(deg F^i_{lm} = a_{lm}\) for every \(1\le i \le s(A)\)? We consider this problem for \(n\ge 4\) and we prove that \(s(A) \le k^{n-3}\) and \(s(A)