Invertible minor assignment: sparse deformations of determinant expansions and their hyperdeterminants

We introduce an algebraic model based on the expansion of the determinant of two matrices, one of which is generic, to check the additivity of Z-valued set functions. Each individual term of the expansion is deformed through a monomial factor in d indeterminates with exponents defined by the set function. A family of sparse polynomials is derived from Grassmann-Plücker relations, and their compatibility is linked to the factorisation of hyperdeterminants in the ring of Laurent polynomials. It is proved that, in broad generality, this deformation returns a determinantal expansion if and only if it is induced by a diagonal matrix of monomials acting as a kernel into the initial determinant expansion, which guarantees the additivity of the set function. The hypotheses underlying this result are tested through the construction of counterexamples, and their implications are explored in terms of complexity reduction, with special attention to permutations of families of subsets.