Minor Invertible Products Assignment and Sparse Hyperdeterminants

We consider an extension of Minor Assignment Problems derived from the determinantal expansion of matrix products, under the condition that the terms of the expansion are units of C(t). This restriction places constraints on the sparsity and the factorization properties of a family of hyperdeterminants derived from Grassmann-Plücker relations. We find minimal conditions guaranteeing that allowed assignments returning a determinantal expansion are the trivial ones, i.e., those induced by the action of a diagonal matrix of Laurent monomials on a pair of constant matrices. Counterexamples are provided when such conditions do not hold. Connections with the characterization of forbidden configurations in a different combinatorial context, as well as potential applications to statistical modeling and choice theory, are also discussed.