The decentralized determinantal assignment problem (DDAP) is defined as the unifying description for the study of pole and zero assignment problems under decentralized output, state feedback (DOF, DSF) and decentralized ‘squaring down’ (DSD), respectively. DDAP is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors. The decentralization characteristic (DC) and the decentralized polynomial Grassmann representative )D — ℝ[s] — GR) of DDAP are defined. The fixed zero polynomial of DDAP is then determined as the zero polynomial of D— ℝ[s]—GR. The canonical D—ℝ[s]—GR, (CD—R[s]—GR) and the decentralized Plücker matrix (DPM) of DDAP are introduced and necessary conditions for arbitrary assignment of the non-fixed zeros are given in terms of the DPM. The family of strongly zero non-assignable (SNA( systems is defined, and for such systems the notion of the fixed zero is extended to that of the ‘almost fixed zero’ (AFZ). An AFZ is defined as the centre of a disc that contains at least one zero of the combinant under any decentralized compensator and as such the AFZs define ‘trapping’ discs for the exact zeros. The general properties and computational aspects of the radii of the ‘trapping’ discs are discussed and a new necessary condition for stabilizability is given in terms of the AFZs. Finally, the results are specialized to the case of pole-zero assignment under DOF (DSF), DSD, respectively. Criteria for fixed and almost fixed modes under DOF (DSF), and fixed and almost fixed zeros under DSD are given, as well as new necessary conditions for decentralized pole-zero assignment.