Let ?=(???) be a non-negative integer ?×? matrix. ? is a homogeneous matrix if ???+???=???+??? for any choice of the four indexes. We ask: If ? is a homogeneous matrix and if ? is a form in ℂ[?1,…??] with ???(?)=trace(?) , what is the least integer, ?(?) , so that ?=????1+⋯+?????(?) , where the ??=(????) are ?×? matrices of forms and ???????=??? for every 1≤?≤?(?) ? We consider this problem for ?≥4 and we prove that ?(?)≤??−3 and ?(?)?−3 in infinitely many cases. However ?(?)=??−3 when the integers in ? are large with respect to ? .
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