A Structural Characterization of Determinantally Equivalent Functions
Let $Λ$ be a set and $\mathbb{F}$ a field. Suppose that $K,Q:Λ^2\to\mathbb{F}$ are two functions such that for any $n\in\mathbb{N}$ and $x_1,x_2,\ldots,x_n\inΛ$, the determinants of matrices $(K(x_i,x_j))_{1\leq i,j\leq n}$ and $(Q(x_i,x_j))_{1\leq i,j\leq n}$ agree. We study to what extent $K$ and $Q$ must be related by two canonical transformations corresponding to diagonal similarity and transposition. In the symmetric case, this relation holds without further assumptions (see [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021]), while in general it fails. In [Harry Sapranidis Mantelos, Determinantally equivalent nonzero functions, Discrete Mathematics, 349(6):115021, 2026], it was shown that the relation remains valid under a natural $2\times 2$ determinantal condition (property $\mathcal{D}$), together with the additional assumption that both functions are nowhere vanishing. We prove that the 'nowhere vanishing' assumption can be removed entirely, and that property $\mathcal{D}$ alone provides the correct and complete structural mechanism governing the problem. In particular, this shows that the nowhere-zero assumption is not intrinsic to the problem, but rather an artefact of the specific method. The proof is entirely combinatorial and avoids linear algebra, relying on an analysis of permutations in the definition of a determinant as cycles in a graph; in particular, it requires new arguments to handle the breakdown of the identities used in \cite{mantelos2026determinantally}, which are crucial to the method therein. In the 'finite $Λ$' case, this also yields a new approach to the classical matrix problem of [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64], thereby revealing an underlying combinatorial structure.