Before revision:  = 0. We show that there is an (n+1)\times(n+1) homeostasis matrix H(I) for which dx_o/dI = 0 if and only if \det(H) = 0. We  use combinatorial matrix theory to factor the polynomial \det(H) and thereby determine a menu of different types of possible homeostasis associated with each digraph G. Specifically, we prove that each factor corresponds to a subnetwork of G. The factors divide into two combinatorially defined classes: structural and appendage.
After revision:  = 0. We show that there is an (n+1)\times(n+1) homeostasis matrix H(I) for which dx_o/dI = 0 if and only if \det(H) = 0. We note that the entries in H are linearized couplings and \det(H) is a homogeneous polynomial of degree n+1 in these entries. We use combinatorial matrix theory to factor the polynomial \det(H) and thereby determine a menu of different types of possible homeostasis associated with each digraph G. Specifically, we prove that each factor corresponds to a subnetwork of G. The factors divide into two combinatorially defined classes: structural and appendage.